Evaluate : $\int_{0}^{\pi} \frac{x}{1 + \sin x} dx$
CBSE CLASS 12 • MATHEMATICS • CHAPTER 07
Integrals
349 distinct questions across 403 board paper set appearances (2022–2026).
Evaluate : $\int_{\pi/2}^{\pi} e^x \left$ \frac{1-\sin x}{1-\cos x} \right$ dx$
(a) Evaluate : $\int_{\frac{\pi}{12}}^{\frac{5\pi}{12}} \frac{dx}{1 + \sqrt{\cot x}}$
OR
(b) Evaluate : $\int_{-\frac{\pi}{6}}^{\frac{\pi}{2}} (\sin |x| + \cos |x|)\, dx$
(a) Evaluate : $\int_{0}^{\pi/2} \sin 2x \cos 3x \, dx$ OR (b) Given $\frac{d}{dx} F(x) = \frac{1}{\sqrt{2x-x^2}}$ and $F(1) = 0$, find $F(x)$.
(a) Find : $\int \cos^3 x \, e^{\log \sin x} \, dx$ OR (b) Find : $\int \frac{1}{5 + 4x - x^2} \, dx$
$\int 2^{x+2} dx$ is equal to :
(a) $2^{x+2} + C$
(b) $2^{x+2} \log 2 + C$
(c) $\frac{2^{x+2}}{\log 2} + C$
(d) $2 \cdot \frac{2^x}{\log 2} + C$
The value of $\int_{0}^{\pi/6} \sin 3x \, dx$ is :
(a) $-\frac{\sqrt{3}}{2}$
(b) $-\frac{1}{3}$
(c) $\frac{\sqrt{3}}{2}$
(d) $\frac{1}{3}$
If $\int \frac{3ax}{b^2 + c^2x^2} \, dx = A \log \left|b^2 + c^2x^2\right| + K$, then the value of A is
(A) $3a$
(B) $\frac{3a}{2b^2}$
(C) $\frac{3a}{b^2c^2}$
(D) $\frac{3a}{2c^2}$
The value of $\int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1}\,dx$ is
(A) $0$
(B) $\log 2$
(C) $2\log 2$
(D) $\frac{1}{2}\log 2$
The integrating factor of differential equation $R \frac{dx}{dy} + Px = Q$ where $P$, $Q$, $R$ are functions of $y$ is
(A) $e^{\int \frac{P}{Q}\,dy}$
(B) $e^{\int P\,dy}$
(C) $e^{\int \frac{P}{R}\,dy}$
(D) $e^{\int \frac{P}{R}\,dx}$
A racing track is build around an elliptical ground whose equation is given by $9x^2 + 16y^2 = 144$. The width of the track is $3\text{ m}$ as shown below :
Based on given information, answer the following questions :
(i) Express $y$ as a function of $x$ from the given equation of ellipse.
(ii) Inte...
(a) Find : $\int \frac{\cos x}{(2+\sin x)(4+\sin x)} \, dx$
OR
(b) Find : $\int \frac{x+3}{x^2+4x+5} \, dx$
If $\int_{0}^{1} \frac{dx}{e^x + e^{-x}} = \tan^{-1} e + k$, then the value of k is : (A) e (B) $\frac{\pi}{4}$ (C) 0 (D) $-\frac{\pi}{4}$
(a) Find :
$\int \frac{x}{(x - 1)(x^2 + 4)} dx$
OR
(b) Evaluate :
$\int_{0}^{1} \frac{x \tan^{-1} x}{(1 + x^2)^{3/2}} dx$
If $\int_{0}^{1} (9x^2 - 4x - 2k)\, dx = 0$, then the value of '$k$' is :
(A) $1$
(B) $0$
(C) $\frac{1}{2}$
(D) $-\frac{1}{2}$
The area of the shaded region bounded by the curves $y^2 = x$, $x = 4$ and the $x$-axis is given by
(A) $\int_{0}^{4} x \, dx$
(B) $\int_{0}^{2} y^2 \, dy$
(C) $2 \int_{0}^{4} \sqrt{x} \, dx$
(D) $\int_{0}^{4} \sqrt{x} \, dx$
$\int \frac{x + 5}{(x + 6)^2} e^x \, dx$ is equal to : (A) $\log (x + 6) + C$ (B) $e^x + C$ (C) $\frac{e^x}{x + 6} + C$ (D) $\frac{-1}{(x + 6)^2} + C$
(a) Find :
$$\int \frac{x^2 + 1}{(x - 1)^2 (x + 3)} \, dx$$
OR
(b) Evaluate :
$$\int_{0}^{\pi/2} \frac{x}{\sin x + \cos x} \, dx$$
$\int \frac{e^{9 \log x} - e^{8 \log x}}{e^{6 \log x} - e^{5 \log x}} \, \text{d}x$ is equal to :
(A) $x + C$
(B) $\frac{x^2}{2} + C$
(C) $\frac{x^4}{4} + C$
(D) $\frac{x^3}{3} + C$
Find : $\int \frac{e^{4x} - 1}{e^{4x} + 1} dx$
(a) Evaluate : $\int_{-2}^{2} \sqrt{\frac{2-x}{2+x}} dx$ OR (b) Find : $\int \frac{1}{x [(\log x)^2 - 3 \log x - 4]} dx$
$\int_{-a}^{a} f(x) \text{ dx} = 0$, if :
(A) $f(-x) = f(x)$
(B) $f(-x) = -f(x)$
(C) $f(a - x) = f(x)$
(D) $f(a - x) = -f(x)$
(a) Evaluate : $\int_{0}^{\frac{\pi}{4}} \frac{x \, dx}{1 + \cos 2x + \sin 2x}$ OR (b) Find : $\int e^{x} \left[ \frac{1}{(1 + x^{2})^{\frac{3}{2}}} + \frac{x}{\sqrt{1 + x^{2}}} \right] dx$.
$\int \frac{1}{x(\log x)^2} dx$ is equal to : (A) $2 \log (\log x) + c$ (B) $-\frac{1}{\log x} + c$ (C) $\frac{(\log x)^3}{3} + c$ (D) $\frac{3}{(\log x)^3} + c$
The value of $\int_{-1}^{1} x|x| \, dx$ is : (A) $\frac{1}{6}$ (B) $\frac{1}{3}$ (C) $-\frac{1}{6}$ (D) $0$
(a) Evaluate : $\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} \, dx$ OR (b) Find : $\int \frac{2x+1}{(x+1)^2 (x-1)} \, dx$
The value of $\int_{\pi/4}^{\pi/2} \cot \theta \csc^2 \theta \ d\theta$ is : (A) $\frac{1}{2}$ (B) $-\frac{1}{2}$ (C) $0$ (D) $-\frac{\pi}{8}$
(a) Find : $\int \cos^3 x e^{\log \sin x} dx$
OR
(b) Find : $\int \frac{1}{5 + 4x - x^2} dx$
(a) Evaluate : $\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx$ OR (b) Find : $\int \frac{2x + 1}{(x + 1)^2 (x - 1)} dx$
(a) Find: $\int \frac{x^3 - 1}{x^3 - x} dx$ OR (b) Evaluate: $\int_{-4}^{0} |x+2| dx$
Evaluate : $\int_{0}^{\frac{\pi}{2}} e^x \sin x \, dx$
Evaluate $\int_{0}^{\pi/4} \log (1 + \tan x) dx$.
Find $\int e^{\cot^{-1} x} \left$ \frac{1-x+x^2}{1+x^2} \right$ dx.$
Evaluate $\int_{\log \sqrt{2}}^{\log \sqrt{3}} \frac{1}{(e^x + e^{-x}) (e^x - e^{-x})} dx$
(a) Find : $\int \frac{\cos x}{\sin 3x} dx$ OR (b) Find : $\int x^{2} \log (x^{2} + 1) dx$
$\int e^{5 \log x} \, dx$ is equal to :
(a) $\frac{x^5}{5} + C$
(b) $\frac{x^6}{6} + C$
(c) $5x^4 + C$
(d) $6x^5 + C$
If $\int_{0}^{a} 3x^2 \, dx = 8$, then the value of 'a' is :
(a) $2$
(b) $4$
(c) $8$
(d) $10$
(a) Evaluate :
$\int_{0}^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx$
OR
(b) Evaluate :
$\int_{1}^{3} (|x - 1| + |x - 2|) dx$
Find : $\int \frac{\cos \theta}{\sqrt{3 - 3 \sin \theta - \cos^2 \theta}} d\theta$
Find : $\int \frac{2x^2 + 1}{x^2(x^2 + 4)} dx$
Find : $\int \frac{\log x - 3}{(\log x)^{4}} dx.$
Evaluate : $\int_{0}^{1} \tan^{-1} x \, dx$
Find : $\int \frac{2x}{x^2 + 3x + 2} \, dx$
Evaluate : $\int_{0}^{1} x (1 - x)^{n} dx$
Find : $\int \frac{dx}{x^2 - 6x + 13}$
Find : $\int \frac{x^2}{(x^2 + 1)(3x^2 + 4)} dx$ OR Evaluate : $\int_{-2}^{1} \sqrt{5 - 4x - x^2} dx$
Evaluate : $\int_{0}^{1} x^2 e^x dx$
(a) Find : $\int \cos x \cdot \tan^{-1}(\sin x) \, dx$ OR (b) Find : $\int \frac{e^x}{(e^x + 1)(e^x + 3)} \, dx$
If $\int \\frac{3ax}{b^2 + c^2 x^2} \, dx = \text{A} \log \left| b^2 + c^2 x^2 \right| + \text{K}$, then the value of $\text{A}$ is
(A) $3a$
(B) $\\frac{3a}{2b^2}$
(C) $\\frac{3a}{b^2c^2}$
(D) $\\frac{3a}{2c^2}$
The value of $\int\limits_{-1}^{1} \\frac{x^3}{x^2 + 2|x| + 1} \, dx$ is
(A) $0$
(B) $\log 2$
(C) $2 \log 2$
(D) $\\frac{1}{2} \log 2$
The integrating factor of differential equation $\text{R} \\frac{dx}{dy} + Px = Q$ where $ ext{P}$, $ ext{Q}$, $ ext{R}$ are functions of $y$ is
(A) $e^{\int \\frac{\text{P}}{\text{Q}} \, dy}$
(B) $e^{\int \text{P} \, dy}$
(C) $e^{\int \\frac{\text{P}}{\text{R}} \, dy}$
(D) $e^{\int \\frac{\text{P}}...
Evaluate : $\int\limits_{0}^{1} x \tan^{-1} x \, \text{d}x$.
(a) Find $\int \sqrt{\\frac{x + 2}{x - 2}} \, \text{d}x$ \
OR \
(b) Find : $\int \\frac{x^2}{(x^2 + 9)(x^2 + 16)} \, \text{d}x$
If $\text{I}_1 = \int\limits_{-\pi/4}^{\pi/4} \\frac{\text{d}x}{1 + \cos 2x}$ and $\text{I}_2 = \int\limits_{-1/2}^{1/2} |x| \, \text{d}x$, then show that $\text{I}_1 - 4\text{I}_2 = 0$.
If $I_1 = \int\limits_{-\pi/4}^{\pi/4} \frac{\text{d}x}{1+\cos 2x}$ and $I_2 = \int\limits_{-1/2}^{1/2} |x| \text{d}x$, then show that $I_1 - 4I_2 = 0$.
Evaluate : $\int_{0}^{1} x \sin^{-1} x \, dx$
(a) Find $\int \sqrt{\frac{x+2}{x-2}} \, dx$
OR
(b) Find : $\int \frac{x^{2}}{\left(x^{2}+9\right)\left(x^{2}+16\right)} \, dx$
Evaluate : $\int_{0}^{1} \log(1 + x^2) \text{d}x$
(a) Find $\int \sqrt{\frac{x + 2}{x - 2}} \text{d}x$\nOR\n(b) Find : $\int \frac{x^2}{(x^2 + 9)(x^2 + 16)} \text{d}x$
If $I_1 = \int_{-\pi/4}^{\pi/4} \frac{\text{d}x}{1 + \cos 2x}$ and $I_2 = \int_{-1/2}^{1/2} |x| \text{d}x$, then show that $I_1 - 4I_2 = 0$.
If $\int_{0}^{2a} \frac{1}{1 + 4x^2} \, dx = \frac{\pi}{6}$, then the value of $a$ is
(A) $\frac{\sqrt{3}}{4}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\sqrt{3}$
(D) $2\sqrt{3}$
Which of the following expressions will give the area of region bounded by the curve $y = x^2$ and line $y = 16$ ?
(A) $\int_{0}^{4} x^2 \, dx$
(B) $2 \int_{0}^{4} x^2 \, dx$
(C) $\int_{0}^{16} \sqrt{y} \, dy$
(D) $2 \int_{0}^{16} \sqrt{y} \, dy$
Find : $\int \frac{x + 2}{\sqrt{9x - x^2}}\, dx$
(ii) Integrate the function obtained in (i) with respect to $x$.
$\int \frac{dx}{1 + \cos x}$ is equal to
(A) $\frac{1}{2} an \frac{x}{2} + C$
(B) $\frac{-1}{2} \cot \frac{x}{2} + C$
(C) $-\cot \frac{x}{2} + C$
(D) $ an \frac{x}{2} + C$
Which of the following expressions will give the area of region bounded by the curve $y = x^2$ and line $y = 16$ ?
(A) $\int\limits_{0}^{4} x^2 \, dx$
(B) $2 \int\limits_{0}^{4} x^2 \, dx$
(C) $\int\limits_{0}^{16} \sqrt{y} \, dy$
(D) $2 \int\limits_{0}^{16} \sqrt{y} \, dy$
Find : $\int \frac{2x + 1}{\sqrt{6x + x^2}} \, dx$
$\int \frac{\text{d}x}{\sec x + \tan x}$ is equal to
(A) $\log \mid \sec x + \tan x \mid + \text{C}$
(B) $\log \mid \sec x - \tan x \mid + \text{C}$
(C) $\log \mid 1 + \cos x \mid + \text{C}$
(D) $\log \mid 1 + \sin x \mid + \text{C}$
Which of the following expressions will give the area of region bounded by the curve $y = x^2$ and line $y = 16$ ?
(A) $\int_{0}^{4} x^2 \text{ d}x$
(B) $2\int_{0}^{4} x^2 \text{ d}x$
(C) $\int_{0}^{16} \sqrt{y} \text{ d}y$
(D) $2\int_{0}^{16} \sqrt{y} \text{ d}y$
Find : $\int \frac{3x - 1}{\sqrt{x^2 - 4x}} \, dx$
$\int \frac{\cos 2x}{\sin^2 x \cos^2 x} \, dx$ is equal to :
(A) $\cot x + \tan x + C$
(B) $-\cot x + \sec x + C$
(C) $-\cot x + \tan x + C$
(D) $-(\tan x + \cot x) + C$
23. Find :
$\int \frac{(x - 4)e^x}{(x - 2)^3} \, dx$
(a) Find : $\int \frac{x}{\sqrt{7 - 6x - x^2}}\,dx$
OR
(b) Find : $\int \frac{2}{(1 - x)(1 + x^2)}\,dx$
Evaluate : $\int_{0}^{\pi} \frac{x \, dx}{1 + \sin x}$
Find : $$\int \frac{\text{d}x}{x^{1/2} + x^{1/3}}$$
Evaluate : $\int \tan^{-1}\left(\frac{1-x}{1+x}\right)\,dx$
(a) Find :
$$\int \frac{dx}{x^{1/2} + x^{1/3}}$$
$\int_{-1}^{1} (1 - |x|) \, \text{dx}$ is equal to :
(A) $2 \int_{0}^{1} (1 + x) \, \text{dx}$
(B) $2 \int_{-1}^{0} (1 + x) \, \text{dx}$
(C) $0$
(D) $2 \int_{-1}^{0} (1 - x) \, \text{dx}$
The area of the shaded region of the circle given below is equal to :
(A) $\int_{1}^{3} \sqrt{9 - y^2} \, \text{dy}$
(B) $2 \int_{1}^{3} \sqrt{9 - y^2} \, \text{dy}$
(C) $\int_{0}^{3} \sqrt{9 - x^2} \, \text{dx}$
(D) $2 \int_{0}^{3} \sqrt{9 - x^2} \, \text{dx}$
(a) Evaluate : $\int \frac{\cos x}{(2 + \sin x)(4 + \sin x)} \text{ d}x$ \
OR \
(b) Evaluate : $\int \frac{x + 3}{x^2 + 4x + 5} \text{ d}x$
(a) Find :
$$\int \frac{\text{d}x}{x^{1/2} + x^{1/3}}$$
OR
(b) Find :
$$\int \tan^{-1}\left(\frac{1 - x}{1 + x}\right) \text{d}x$$
Evaluate :
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos^2 x}{2^x + 1} \text{d}x$$
$\int_{-1}^{1} (1 - |x|) \, dx$ is equal to :
The area of the shaded region of the circle given below is equal to :
(A) $\int_{1}^{3} \sqrt{9-y^2} \, dy$
(B) $2 \int_{1}^{3} \sqrt{9-y^2} \, dy$
(C) $\int_{0}^{3} \sqrt{9-x^2} \, dx$
(D) $2 \int_{0}^{3} \sqrt{9-x^2} \, dx$
Evaluate: $\int \frac{\text{dx}}{x^{1/2} + x^{1/3}} \text{ OR } \int \tan^{-1}\left(\frac{1-x}{1+x}\right) \text{dx}$
Evaluate : $\int_{0}^{\frac{\pi}{2}} \frac{\sin^{2} x}{1 + \sin 2x} dx$
(a) Find : $\int \frac{\cos x}{(2 + \sin x)(4 + \sin x)} dx$ OR (b) Find : $\int \frac{x + 3}{x^{2} + 4x + 5} dx$
If $\int_{0}^{1} \frac{\text{dx}}{e^x + e^{-x}} = \tan^{-1} e + k$, then the value of $k$ is : (A) $e$ (B) $\frac{\pi}{4}$ (C) $0$ (D) $-\frac{\pi}{4}$
(a) Find :
$\int \frac{x}{(x - 1)(x^2 + 4)} dx$
OR
Evaluate : $\int_{0}^{1} \frac{x \tan^{-1} x}{(1 + x^2)^{3/2}} dx$
$\int \frac{1}{\sqrt{1 + \cos 2x}} dx$ is equal to : (A) $\log \cos x + C$ (B) $\frac{1}{\sqrt{2}} \log |\sec x + \tan x| + C$ (C) $\frac{1}{\sqrt{2}} \log |\sec x - \tan x| + C$ (D) $\log \sin 2x + C$
The value of $\int_{-5}^{-1} \frac{1}{x} dx$ is equal to : (A) $-\log 5$ (B) $x^6$ (C) $\log (-5)$ (D) $x^{-6}$
Find : $\int \frac{x-\sin x}{1-\cos x} dx$
Evaluate : $\int_{0}^{2} \frac{1}{\sqrt{x^2+2x+3}} dx$
$\int \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}} \, dx$ is equal to : (A) $\log |\sec^2 x| + C$ (B) $\log |\cos x| + C$ (C) $\log |1 + \cos 2x| + C$ (D) $\log |\sec x| + C$
The value of $\int_{0}^{1} \frac{1}{3x - 4} \, dx$ is : (A) $\frac{1}{3} \log 4$ (B) $-\frac{1}{3} \log 4$ (C) $\log (-4)$ (D) $\log 4$
(a) Find : $\int \frac{x - \sin x}{1 - \cos x} \, dx$ OR (b) Evaluate : $\int_{0}^{2} \frac{1}{\sqrt{x^2 + 2x + 3}} \, dx$
$\int \sqrt{\frac{1 + \cos \text{x}}{1 - \cos \text{x}}} \text{ dx}$ is equal to :
(A) $2 \log | \sin \frac{\text{x}}{2} | + \text{C}$
(B) $\frac{1}{2} \log | \sin 2\text{x} | + \text{C}$
(C) $\log | 1 - \cos 2\text{x} | + \text{C}$
(D) $\log | 1 + \cos 2\text{x} | + \text{C}$
If $\int_{0}^{1} (6\text{x}^{2} - 4\text{x} + \text{k}) \text{ dx} = 0$, then the value of $\text{k}$ is :
(A) $1$
(B) $0$
(C) $2$
(D) $- 1$
(a) Find : $\int \frac{x - \sin x}{1 - \cos x} dx$ OR (b) Evaluate : $\int_{0}^{2} \frac{1}{\sqrt{x^2 + 2x + 3}} dx$
$\int x \sin^3 x^2 \cos x^2 \text{ dx}$ is equal to :
(A) $\frac{1}{8} \sin^4 x^2 + C$
(B) $\frac{1}{4} \sin^4 x^2 + C$
(C) $\frac{1}{2} \sin^4 x^2 + C$
(D) $\frac{1}{8} \sin^4 x + C$
Find :
$$\int \sec^3 (3x + 2) \tan (3x + 2) \, \text{dx}$$
(a) Find : $\int \frac{e^x}{\sqrt{5 - 4e^x - e^{2x}}} \, \text{dx}$ \
OR \
(b) Find : $\int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, \text{dx}$
Evaluate : $\int_{1}^{3} (|x - 1| + |x - 2| + |x - 3|) \, \text{dx}$
$\int \left[ \frac{1}{\log x} - \frac{1}{(\log x)^2} \right] dx$ is equal to :
(A) $\frac{x}{\log x} + C$
(B) $x^2 \log x + C$
(C) $\frac{1}{\log x} + C$
(D) $\frac{-1}{(\log x)^2} + C$
If $\int_{0}^{4} \frac{1}{x + \sqrt{x}} dx = k \log 3$, then the value of $k$ is :
(A) $4$
(B) $-2$
(C) $-4$
(D) $2$
Evaluate :
$$\int_{\pi/6}^{\pi/3} \frac{\sin x + \cos x}{\sqrt{\sin 2x}}\, \mathrm{dx}$$
If $\int_{0}^{1} (9x^2 - 4x - 2k) \, \text{dx} = 0$, then the value of '$k$' is :
(A) $1$
(B) $0$
(C) $\frac{1}{2}$
(D) $-\frac{1}{2}$
$\int [\sin (\log x) + \cos (\log x)] \, dx$ is equal to :
(A) $-\cos (\log x) + C$
(B) $\frac{1}{\cos x} + \frac{1}{\sin x} + C$
(C) $\sin (\log x) + C$
(D) $x \sin (\log x) + C$
If $\int_{0}^{4} \frac{1}{x + \sqrt{x}} \, dx = k \log 3$, then the value of $k$ is :
(A) $4$
(B) $-2$
(C) $-4$
(D) $2$
Evaluate :
$$\int_{0}^{\pi/2} \frac{\sin^2 x}{1 + \sin x \cos x} \, \text{dx}$$
If $\int_{0}^{4} \frac{1}{x + \sqrt{x}}\, \mathrm{dx} = k\log 3$, then the value of k is :
(A) $4$
(B) $-2$
(C) $-4$
(D) $2$
$\int e^x \left(\frac{2 + \sin 2x}{1 + \cos 2x}\right) dx$ is equal to :
(A) $\log \cos 2x + C$
(B) $\sec^2 2x + C$
(C) $e^x \tan x + C$
(D) $e^x \cot x + C$
Evaluate : $I = \int_{-\pi}^{\pi} \frac{2x(1 + \sin x)}{1 + \cos^2 x} \, \text{dx}$
If $f(a+b-x) = f(x)$, then $\int_{a}^{b} x f(x) dx$ is equal to
(A) $\frac{a+b}{2} \int_{a}^{b} f(b-x) dx$
(B) $\frac{a+b}{2} \int_{a}^{b} f(a-x) dx$
(C) $\frac{b-a}{2} \int_{a}^{b} f(x)\ dx$
(D) $\frac{a+b}{2} \int_{a}^{b} f(x) dx$
If $\int x^3 \sin^4(x^4) \cos(x^4) dx = a \sin^5(x^4) + C$, then $a$ is equal to
(A) $-\frac{1}{10}$
(B) $\frac{1}{20}$
(C) $\frac{1}{4}$
(D) $\frac{1}{5}$
Find: $\int \frac{(x-3)}{(x-1)^3} e^x \, dx$
Sketch the graph $y = |x + 1|$. Evaluate $\int_{-4}^2 |x + 1|dx$. What does the value of this integral represent on the graph?
For Visually Impaired:
Define the function $y = |x + 1|$. Evaluate $\int_{-4}^2 |x + 1|dx$. What does the value of this integral represent?
Evaluate: $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} \, dx$
Find $\int \frac{$3 \sin \theta - 2$ \cos \theta}{5 - \cos^2 \theta - 4 \sin \theta} \, d\theta$
(a) Find : $\int \frac{\text{d}x}{(x+1)^2(x^2+1)}$
OR
(b) Solve the differential equation : $\frac{\text{d}y}{\text{d}x} = e^{x-y} + x^2e^{-y}$
The value of $\int_{-2}^{2} \sin^5 x \cos x \, dx$ is :
(A) $\frac{64}{3}$
(B) $0$
(C) $2 \sin^6 2$
(D) $\sin^6 (-2) - \sin^6 2$
(a) Find : $\int \sin^3 x \cdot \cos^4 x \, dx$
(b) Evaluate : $\int_{-2}^{1} |x^3 - x| \, dx$
$\int \frac{3 \cos \sqrt{x}}{\sqrt{x}} \, \text{d}x$ is equal to :
(A) $-6 \sin \sqrt{x} + C$
(B) $-6 \cos \sqrt{x} + C$
(C) $6 \cos \sqrt{x} + C$
(D) $6 \sin \sqrt{x} + C$
(a) Find :
$\int \sqrt{4x^2-4x+10}\, \text{d}x$
OR
(b) Evaluate :
$\int_{0}^{\pi} \frac{x\sin x}{1+\cos^2 x}\, \text{d}x$
$\int \frac{\tan^2\sqrt{x}}{\sqrt{x}} \, dx$ is equal to :
(A) $\sec\sqrt{x} + C$
(B) $2\sqrt{x} \tan x - x + C$
(C) $2(\tan\sqrt{x} - \sqrt{x}) + C$
(D) $2\tan\sqrt{x} - x + C$
(a) Evaluate :
$I = \int_{2}^{4} (|x - 2| + |x - 3| + |x - 4|)\,dx$
OR
(b) Find :
$\int \frac{dx}{(x + 2)(x^2 + 1)}$
$\int \frac{\cot\sqrt{x}}{\sqrt{x}} \, dx$ is equal to :
(A) $\log \sin\sqrt{x} + C$
(B) $\frac{\log \sin\sqrt{x}}{2\sqrt{x}} + C$
(C) $2\log \sin\sqrt{x} + C$
(D) $\frac{\log \sin\sqrt{x}}{\sqrt{x}} + C$
(a) Find : $\int \sqrt{4\text{x}^2 - 4\text{x} + 10} \text{ dx}\n\textbf{OR}\n(b) \text{Evaluate : } \int_0^{\pi} \frac{\text{x}\sin\text{x}}{1 + \cos^2\text{x}} \text{ dx}$
$\int_a^b \text{f}(x) \text{ d}x$ is equal to :
(A) $\int_a^b \text{f}(a-x) \text{ d}x$
(B) $\int_a^b \text{f}(a+b-x) \text{ d}x$
(C) $\int_a^b \text{f}(x - (a+b)) \text{ d}x$
(D) $\int_a^b \text{f}((a-x) + (b-x)) \text{ d}x$
(a) Evaluate : $\int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16 \sin 2x} \, \text{dx}$ \
OR \
(b) Evaluate : $\int_{0}^{\frac{\pi}{2}} \sin 2x \tan^{-1}(\sin x) \, \text{dx}$
(a) Find :
$\int x \sqrt{1 + 2x} dx$
OR
(b) Evaluate :
$\int_{0}^{\frac{\pi^2}{4}} \frac{\sin \sqrt{x}}{\sqrt{x}} dx$
(a) Find :
$$\int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx$$
OR
(b) Evaluate :
$$\int_{1}^{3} (|x - 1| + |x - 2| + |x - 3|) \, dx$$
$\int_{-1}^{1} \frac{|x|}{x} \, dx$, $x \neq 0$ is equal to
(A) $-1$
(B) $0$
(C) $1$
(D) $2$
If $\int \frac{2^{\frac{1}{x}}}{x^2} dx = k \cdot 2^{\frac{1}{x}} + C$, then $k$ is equal to
(A) $\frac{-1}{\log 2}$
(B) $-\log 2$
(C) $-1$
(D) $\frac{1}{2}$
(a) Find : $\int \frac{x + \sin x}{1 + \cos x} \text{d}x$
OR
(b) Evaluate : $\int_{0}^{\frac{\pi}{4}} \frac{\text{d}x}{\cos^3 x \sqrt{2 \sin 2x}}$
$\int \frac{1 - 2\sin x}{\cos^2 x} \, dx$ is equal to :
(A) $\tan x - 2\sec x + C$
(B) $-\tan x + 2\sec x + C$
(C) $-\tan x - 2\sec x + C$
(D) $\tan x + 2\sec x + C$
If $\int \frac{\frac{1}{2^x}}{x^2} \text{dx} = k \cdot 2^{\frac{1}{x}} + C$, then $k$ is equal to
(A) $\frac{-1}{\log 2}$
(B) $-\log 2$
(C) $-1$
(D) $\frac{1}{2}$
(a) Find : $\int \frac{\cos 2x}{(\sin x + \cos x)^2} dx$
OR
(b) Evaluate : $\int_{0}^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} dx$
If $\int_{0}^{1} \frac{e^{x}}{1+x} \, dx = \alpha$, then $\int_{0}^{1} \frac{e^{x}}{(1+x)^{2}} \, dx$ is equal to
(A) $\alpha - 1 + \frac{e}{2}$
(B) $\alpha + 1 - \frac{e}{2}$
(C) $\alpha - 1 - \frac{e}{2}$
(D) $\alpha + 1 + \frac{e}{2}$
If $\int \frac{2^{x}}{x^{2}} \, dx = k \cdot 2^{x} + C$, then k is equal to
(A) $\frac{-1}{\log 2}$
(B) $-\log 2$
(C) $-1$
(D) $\frac{1}{2}$
The area of the shaded region bounded by the curves $y^2 = x$, $x = 4$ and the $x$-axis is given by
(A) $\int_{0}^{4} x \text{ dx}$
(B) $\int_{0}^{2} y^2 \text{ dy}$
(C) $2 \int_{0}^{4} \sqrt{x} \text{ dx}$
(D) $\int_{0}^{4} \sqrt{x} \text{ dx}$
(a) Find : $\int \frac{2x - 1}{(x - 1)(x + 2)(x - 3)}\,dx$\nOR\n(b) Evaluate : $\int_{0}^{5} (|\,x - 1\,| + |\,x - 2\,| + |\,x - 5\,|)\,dx$
Find : $\int \sin^{-1} \sqrt{\frac{x}{a + x}} \, dx$.
If $f(2a - x) = f(x)$, then $\int_{0}^{2a} f(x) \, dx$ is
(A) $\int_{0}^{2a} f\left$ \frac{x}{2} \right$ \, dx$
(B) $\int_{0}^{a} f(x) \, dx$
(C) $2 \int_{a}^{0} f(x) \, dx$
(D) $2 \int_{0}^{a} f(x) \, dx$
The area of the shaded region (figure) represented by the curves $y = x^2$, $0 \le x \le 2$ and $y$-axis is given by
(A) $\int_{0}^{2} x^2 \, \text{d}x$
(B) $\int_{0}^{2} \sqrt{y} \, \text{d}y$
(C) $\int_{0}^{4} x^2 \, \text{d}x$
(D) $\int_{0}^{4} \sqrt{y} \, \text{d}y$
Evaluate : $\int_{0}^{\frac{\pi}{4}} \sqrt{1 + \sin 2x} \text{ dx}$
Find: $\int \frac{1}{x} \sqrt{\frac{x+a}{x-a}} \, dx$.
Find: $\int \frac{x^2+x+1}{(x+2)(x^2+1)} \, dx$.
$\int \frac{dx}{\sin^2 x \cos^2 x}$ is equal to
(A) $\tan x + \cot x + C$
(B) $(\tan x + \cot x)^2 + C$
(C) $\tan x - \cot x + C$
(D) $(\tan x - \cot x)^2 + C$
Evaluate : $\int_{0}^{\pi} \frac{\sin 2px}{\sin x} dx, p \in \mathbb{N}$.
$f$ and $g$ are continuous functions on interval $[a, b]$. Given that $f(a - x) = f(x)$
and $g(x) + g(a - x) = a$, show that $\int\limits_{0}^{a} f(x) \, g(x) \, \text{d}x = \frac{a}{2} \int\limits_{0}^{a} f(x) \, \text{d}x$.
Find : $\int \frac{5x}{(x+1)(x^2+9)}\text{d}x$.
$\int e^x \cdot \frac{x}{(1+x)^2} \, dx$ is equal to
(A) $e^x \cdot \frac{x}{1+x} + C$
(B) $e^x \cdot \frac{1}{1+x} + C$
(C) $e^x \cdot \frac{1}{x} + C$
(D) $e^x \cdot \frac{1}{(1+x)^2} + C$
(a) Find : $\int \frac{x^2 - x + 1}{(x - 1)(x^2 + 1)} dx$
OR
(b) Evaluate : $\int_{1}^{4} (|x| + |3 - x|) dx$
$\int \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha} dx$ is equal to :
(A) $2(\sin x + x \cos \alpha) + C$
(B) $2(\sin x - x \cos \alpha) + C$
(C) $2(\sin x + 2x \cos \alpha) + C$
(D) $2(\sin x + \sin \alpha) + C$
(a) Find :
$$\int \frac{\cos x}{(4+\sin^2 x)(5-4\cos^2 x)} \, \mathrm{dx}$$
OR
(b) Evaluate :
$$\int_{0}^{\pi} \frac{\mathrm{dx}}{a^2 \cos^2 x + b^2 \sin^2 x}$$
$\int e^x (\cos x - \sin x) dx$ is equal to :
(A) $e^x \sin x + C$
(B) $-e^x \sin x + C$
(C) $-e^x \cos x + C$
(D) $e^x \cos x + C$
Find : $\int \frac{\cos x \, dx}{1 + \cos x + \sin x}$
(a) Evaluate: $\int_{0}^{\pi/4} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} dx$ OR (b) Find: $\int \frac{\sqrt{x^2+1} [\log(x^2+1) - 2 \log x]}{x^2} dx$
If \int e^{-3 \log x} dx = f(x) + C, then f(x) is : (A) e^{-3 \log x} (B) e^{\log\left(\frac{1}{x^3}\right)} (C) \frac{-1}{2x^2} (D) \frac{-1}{4x^4}
Find : $\int \frac{\sqrt{x}}{1+\sqrt{x^{3/2}}} dx$
(a) Evaluate :
$$\int_{0}^{3/2} |x \cos \pi x| \, \text{dx}$$
(b) Find :
$$\int \frac{\text{dx}}{\sin x + \sin 2x}$$
$\int \sqrt{1+\sin x} \, dx$ is equal to : (A) $2\left$-\sin \frac{x}{2} + \cos \frac{x}{2}\right$ + C$ (B) $2\left$\sin \frac{x}{2} - \cos \frac{x}{2}\right$ + C$ (C) $-2\left$\sin \frac{x}{2} + \cos \frac{x}{2}\right$ + C$ (D) $2\left$\sin \frac{x}{2} + \cos \frac{x}{2}\right$ + C$
$\int_{0}^{\pi/2} \cos x \cdot e^{\sin x} \, dx$ is equal to : (A) $0$ (B) $1 - e$ (C) $e - 1$ (D) $e$
(a) Find : $\int \frac{2x}{(x^2+3)(x^2-5)} \, dx$ \
OR \
(b) Evaluate : $\int_{1}^{4} (|x-2| + |x-4|) \, dx$
$\int \frac{\cos 2x}{\sin^2 x \cos^2 x} \, dx$ is equal to :
(A) $\cot x + \tan x + C$
(B) $-(\cot x + \tan x) + C$
(C) $-\cot x + \tan x + C$
(D) $\cot x - \tan x + C$
If $\int_{0}^{a} \frac{1}{1 + 4x^2} \, dx = \frac{\pi}{8}$, then the value of 'a' is :
(A) $\frac{1}{4}$
(B) $\frac{1}{2}$
(C) $\frac{1}{8}$
(D) 4
(a) Find : $\int \frac{2x}{(x^2+3)(x^2-5)} \, \text{d}x$ \
OR \
(b) Evaluate : $\int_{1}^{4} (|x-2| + |x-4|) \, \text{d}x$
(a) Find :
$$\int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, \text{d}x$$
OR
(b) Evaluate :
$$\int_{1}^{4} (|x - 2| + |x - 4|) \, \text{d}x$$
(a) Find : $\int \frac{x^2 + 1}{(x - 1)^2 (x + 3)}\, dx$ \
OR \
(b) Evaluate : $\int_{0}^{\pi/2} \frac{x}{\sin x + \cos x}\, dx$
$\int \frac{x+5}{(x+6)^2} e^x \, \text{dx}$ is equal to :
(A) $\log (x+6) + C$
(B) $e^x + C$
(C) $\frac{e^x}{x+6} + C$
(D) $\frac{-1}{(x+6)^2} + C$
$\int \frac{e^{9\log x} - e^{8\log x}}{e^{6\log x} - e^{5\log x}} \, dx$ is equal to :
(A) $x + C$
(B) $\frac{x^2}{2} + C$
(C) $\frac{x^4}{4} + C$
(D) $\frac{x^3}{3} + C$
For a function $f(x)$, which of the following holds true ?
(A) $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx$
(B) $\int_{-a}^{a} f(x) \, dx = 0$, if $f$ is an even function
(C) $\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$, if $f$ is an odd function
(D) $\int_{0}^{2a} f(x) \, d...
(a) Find : $\int \frac{x^2 + 1}{(x^2 + 2)(2x^2 + 1)} dx$
OR
(b) Evaluate : $\int_{0}^{\pi} \frac{x \tan x}{\sec x + \tan x} dx$
For a function $f(x)$, which of the following holds true ?
(A) $\int_{a}^{b} f(x) \text{dx} = \int_{a}^{b} f(a + b - x) \text{dx}$
(B) $\int_{-a}^{a} f(x) \text{dx} = 0$, if $f$ is an even function
(C) $\int_{-a}^{a} f(x) \text{dx} = 2 \int_{0}^{a} f(x) \text{dx}$, if $f$ is an odd function
(D) $\in...
(a) Find :
$$\int \frac{x}{(x - 1)(x^2 + 4)} \, dx$$
OR
(b) Evaluate :
$$\int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx$$
For a function $f(x)$, which of the following holds true ?
(A) $\int\limits_{a}^{b} f(x)\,dx = \int\limits_{a}^{b} f(a + b - x)\,dx$
(B) $\int\limits_{-a}^{a} f(x)\,dx = 0$, if $f$ is an even function
(C) $\int\limits_{-a}^{a} f(x)\,dx = 2 \int\limits_{0}^{a} f(x)\,dx$, if $f$ is an odd function
(D)...
(a) Find : $\int \frac{3x + 1}{(x - 2)^2 (x + 2)} dx$
(b) Evaluate : $\int_{0}^{\pi/2} \frac{x}{\cos x + \sin x} dx$
If $f(x)$ is an odd function, then $\int_{-\pi/2}^{\pi/2} f(x) \cos^3 x \, dx$ equals :
(A) $2 \int_0^{\pi/2} f(x) \cos^3 x \, dx$
(B) $0$
(C) $2 \int_0^{\pi/2} f(x) \, dx$
(D) $2 \int_0^{\pi/2} \cos^3 x \, dx$
(a) Find :
$$\int x\sqrt{1 + 2x} \text{ d}x$$
OR
(b) Evaluate :
$$\int_{0}^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \text{ d}x$$
(a) Find :
$$\int \frac{x^2}{(x^2 + 4)(x^2 + 9)}\ \mathrm{d}x$$
OR
(b) Evaluate :
$$\int_{1}^{3} (|x - 1| + |x - 2| + |x - 3|)\ \mathrm{d}x$$
(a) Find : $\int \frac{(3 \cos \text{x} - 2) \sin \text{x}}{5 - \sin^2 \text{x} - 4 \cos \text{x}} \text{dx}$
OR
(b) Evaluate : $\int_{-2}^{2} \frac{\text{x}^3 + |\text{x}| + 1}{\text{x}^2 + 4 |\text{x}| + 4} \text{dx}$
$Q.12. \int \frac{dx}{x^3(1+x^4)^{\frac{1}{2}}}$ equals
(A) $-\frac{1}{2x^2} \sqrt{1+x^4} + c$
(B) $\frac{1}{2x} \sqrt{1+x^4} + c$
(C) $-\frac{1}{4x} \sqrt{1+x^4} + c$
(D) $\frac{1}{4x^2} \sqrt{1+x^4} + c$
$Q.13. \int_{0}^{2\pi} \operatorname{cosec}^7 x \, dx =$
(A) 0
(B) 1
(C) 4
(D) $2\pi$
Evaluate: $\int \left\{ \frac{1}{\log x} - \frac{1}{(\log x)^2} \right\} dx$; (where $x > 1$).
Evaluate: $\int_{0}^{1} x(1-x)^n dx$; (where $n \in \mathbb{N}$).
$\int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} \text{ dx}$ is equal to :
(A) $\pi$
(B) $\text{Zero (0)}$
(C) $\int_{0}^{\pi/2} \frac{2\sin x}{1 + \sin x \cos x} \text{ dx}$
(D) $\frac{\pi^2}{4}$
Find : $\int x^2 \cdot \sin^{-1} (x^{3/2}) dx$
If $\int_{0}^{2} 2 e^{2x} dx = \int_{0}^{a} e^{x} dx$, the value of 'a' is : (A) 1 (B) 2 (C) 4 (D) $\frac{1}{2}$
Evaluate : $\int_{0}^{a^3} \frac{x^2}{x^6 + a^6} dx$
Find : $\int x^2 . \sin^{-1}(x^{3/2}) dx$
Evaluate : $\int_{-2}^{2} \sqrt{\frac{2 - x}{2 + x}} dx$
Find : $\int \frac{1}{x [(\log x)^2 - 3 \log x - 4]} dx$
Find : $\int x^2 \log (x^2 - 1) dx$
$\int \frac{\cos 2x}{\sin^2 x \cos^2 x} dx$ is equal to :
(A) $\cot x + \tan x + c$
(B) $-\cot x + \tan x + c$
(C) $\cot x - \tan x + c$
(D) $-\cot x - \tan x + c$
Find : $\int \csc^3(3x + 1) \cot(3x + 1) \, dx$
Find : $\int \frac{2x}{(x^2 + 1)(x^2 + 3)} \, dx$
(a) Evaluate : $\int_{-6}^{6} |x + 2| \, dx$ OR (b) Find : $\int \left(\frac{4 - x}{x^5}\right) e^x \, dx$
Find : $\int \frac{x^2+1}{(x^2+2)(x^2+4)} dx$
(a) Find : $\int \frac{2+\sin 2x}{1+\cos 2x} e^x dx$
OR
(b) Evaluate : $\int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} dx$
If $g(x)$ is a continuous function satisfying $g(-x) = -g(x)$, then $\int_{0}^{2a} g(x) dx$ is equal to :
(A) 0
(B) $2 \int_{0}^{a} g(x) dx$
(C) $\int_{-a}^{a} g(x) dx$
(D) $-\int_{-2a}^{0} g(x) dx$
Find : $\int \frac{x^{-1}}{(\log x)^2 - 5\log x + 4} dx$
(a) Find : $\int \frac{2 + \sin 2x}{1 + \cos 2x} e^x dx$ \n\nOR \n\n(b) Evaluate : $\int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} dx$
Find : $\int \frac{\sqrt{x}}{(x + 1)(x - 1)} dx$
Find : $\int \frac{2 + \sin 2x}{1 + \cos 2x} e^x dx$
Evaluate : $\int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} dx$
The value of $\int_{0}^{3} \frac{dx}{\sqrt{9-x^2}}$ is :
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{2}$
(D) $\frac{\pi}{18}$
Find : $\int \frac{1}{x(x^{2} - 1)} \, dx$.
Find : $\int \frac{3x + 5}{\sqrt{x^2 + 2x + 4}} dx$
The value of \int_{-1}^{1} |x| dx is : (A) -2 (B) -1 (C) 1 (D) 2
Evaluate: $\int_{\frac{-1}{2}}^{\frac{1}{2}} \cos x \cdot \log \left$ \frac{1 + x}{1 - x} \right$ dx$
Find : $\int \frac{2x + 3}{x^2(x + 3)} dx$
Evaluate: $\int_{0}^{\frac{\pi}{4}} \frac{x \text{ d}x}{1 + \cos 2x + \sin 2x}$
Find: $\int e^{x} \left[ \frac{1}{(1 + x^{2})^{\frac{3}{2}}} + \frac{x}{\sqrt{1 + x^{2}}} \right] \text{d}x$
The value of $\int_{0}^{\pi} \tan^{2} \left$\frac{\theta}{3}\right$ d\theta$ is :
(A) $\pi + \sqrt{3}$
(B) $3\sqrt{3} - \pi$
(C) $\sqrt{3} - \pi$
(D) $\pi - \sqrt{3}$
Find : $\int \frac{2x}{(x^2+1)(x^2-4)} \, dx$.
Find : $\int \sec^3 \theta \, d\theta$
Evaluate : $\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} \text{dx}$
Find : $\int \frac{2x + 1}{(x + 1)^{2}(x - 1)} \text{dx}$
If $\int_{-2}^{3} x^2 \, dx = k \int_{0}^{2} x^2 \, dx + \int_{2}^{3} x^2 \, dx$, then the value of k is :
(A) 2
(B) 1
(C) 0
(D) $\frac{1}{2}$
The value of $\int_{1}^{e} \log x \, dx$ is :
(A) 0
(B) 1
(C) e
(D) e $\log$ e
If $\int_{-2}^{3} x^2 dx = k \int_{0}^{2} x^2 dx + \int_{2}^{3} x^2 dx$, then the value of k is : (A) 2 (B) 1 (C) 0 (D) $\frac{1}{2}$
The value of $\int_{1}^{e} \log x dx$ is : (A) 0 (B) 1 (C) e (D) $e \log e$
Find : $\int \cos^3 x \ e^{\log \sin x} \ dx$
Find : $\int \frac{1}{5 + 4x - x^2} \ dx$
Evaluate : $\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx$
Find : $\int \frac{2x+1}{(x+1)^2(x-1)} dx$
$\int 2^{2x} \cdot 3^x dx$ is equal to :
(A) $\frac{12^x}{\log 12} + C$
(B) $\frac{2^{2x} \cdot 3^x}{\log 2 \cdot \log 3} + C$
(C) $\frac{4 \cdot 6^x}{\log 6} + C$
(D) $(12)^x \cdot \log 12 + C$
(b) Find : $\int \frac{x^3}{x^4 + 3x^2 + 2} dx$
$\int_{-\pi/4}^{\pi/4} x^3 \cos^2 x \, dx$ is equal to : (A) 0 (B) $-1$ (C) 1 (D) 2
$\int \frac{x - 3}{(x - 1)^3} e^x \, dx$ is equal to : (A) $\frac{2 e^x}{(x - 1)^3} + C$ (B) $\frac{-2 e^x}{(x - 1)^2} + C$ (C) $\frac{e^x}{(x - 1)} + C$ (D) $\frac{e^x}{(x - 1)^2} + C$
(a) Find: $\int \frac{dx}{\cos x \sqrt{\cos 2x}}$ OR (b) Find: $\int \frac{5x - 3}{\sqrt{1 + 4x - 2x^2}} dx$
$\int \frac{x - 3}{(x - 1)^3} e^x dx$ is equal to :
(A) $\frac{2 e^x}{(x - 1)^3} + C$
(B) $\frac{-2 e^x}{(x - 1)^2} + C$
(C) $\frac{e^x}{(x - 1)} + C$
(D) $\frac{e^x}{(x - 1)^2} + C$
$\int_{\pi/6}^{\pi/3} \log(\tan x) \, dx$ is equal to : (A) $\frac{\pi}{2}$ (B) $\frac{\pi}{4}$ (C) 0 (D) $\frac{\pi}{12}$
(a) Find : $\int \frac{dx}{\cos x \sqrt{\cos 2x}}$
Find : $\int \frac{5x - 3}{\sqrt{1 + 4x - 2x^2}} \text{dx}$
$\int_{0}^{\frac{\pi}{6}} \sec^{2}$x - \frac{\pi}{6}$ dx$ is equal to : (a) $\frac{1}{\sqrt{3}}$ (b) $-\frac{1}{\sqrt{3}}$ (c) $\sqrt{3}$ (d) $-\sqrt{3}$
Evaluate : $\int_{0}^{\frac{\pi}{2}} [\log (\sin x) - \log (2 \cos x)] dx$.
Find : $\int \frac{1}{\sqrt{x}(\sqrt{x} + 1)(\sqrt{x} + 2)} dx$.
(a) Evaluate : $\int_{0}^{\frac{\pi}{2}} e^{x} \sin x \, dx$
OR
(b) Find : $\int \frac{1}{\cos(x - a) \cos(x - b)} \, dx$
$\int \frac{1 + \tan x}{1 - \tan x} \text{dx}$ is equal to :
(a) $\sec^2 \left$ \frac{\pi}{4} + x \right$ + C$
(b) $\sec^2 \left$ \frac{\pi}{4} - x \right$ + C$
(c) $\log \left| \sec \left$ \frac{\pi}{4} + x \right$ \right| + C$
(d) $\log \left| \sec \left$ \frac{\pi}{4} - x \right$ \right| + C$
$\int_{0}^{\frac{\pi}{6}} \sec^2 \left$ x - \frac{\pi}{6} \right$ \text{dx}$ is equal to :
(a) $\frac{1}{\sqrt{3}}$
(b) $-\frac{1}{\sqrt{3}}$
(c) $\sqrt{3}$
(d) $-\sqrt{3}$
Find : $\int \frac{1}{\cos(x - a) \cos(x - b)} \, dx$
Evaluate : $\int_{0}^{\frac{\pi}{2}} |\sin x - \cos x| dx$
Find : $\int \frac{1}{\sqrt{x}(\sqrt{x} + 1)(\sqrt{x} + 2)} dx$
$\int_{0}^{\frac{\pi}{6}} \sec^{2}$x - \frac{\pi}{6}$ \text{dx}$ is equal to :
(a) $\frac{1}{\sqrt{3}}$
(b) $- \frac{1}{\sqrt{3}}$
(c) $\sqrt{3}$
(d) $- \sqrt{3}$
Find : $\int \frac{1}{\cos(x - a) \cos(x - b)} \text{ dx}$
Evaluate : $\int_{0}^{\frac{\pi}{2}} [\log(\sin x) - \log(2 \cos x)] \text{ dx}$
Find : $\int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \text{ dx}$
For any integer $n$, the value of $\int_{-\pi}^{\pi} e^{\cos^2x} \sin^3(2n + 1)x \, dx$ is (a) -1 (b) 0 (c) 1 (d) 2
Evaluate : $\int_{-1}^{1} \log\left(\frac{2-x}{2+x}\right)dx$.
Find : $\int \frac{2x^2+3}{x^2(x^2+9)}dx; x \neq 0$.
(iii) Find $P(X>2)$. Find : $\int \sqrt{\frac{x}{1-x^{3}}} dx; x \in (0,1)$. OR Evaluate: $\int_{0}^{\frac{\pi}{4}} \log(1+\tan x)dx$.
Find $\int \frac{dx}{\sqrt{\sin^3x \cos (x - \alpha)}}$.
Evaluate $\int_{\log \sqrt{2}}^{\log \sqrt{3}} \frac{1}{(e^x + e^{-x})(e^x - e^{-x})} \text{dx}$
Evaluate $\int_{-1}^{1} |x^4 - x| dx$.
Find $\int \frac{\sin^{-1}x}{(1 - x^2)^{3/2}} dx$.
Find $\int e^x \left$ \frac{1 - \sin x}{1 - \cos x} \right$ dx$
(a) Find $\int \frac{x + 2}{\sqrt{x^{2} - 4x - 5}} dx$. OR (b) Evaluate $\int_{-a}^{a} f(x) dx$, where $f(x) = \frac{9^{x}}{1 + 9^{x}}$.
If $\int_{0}^{2\pi} \cos^{2}x \, dx = k \int_{0}^{\pi/2} \cos^{2}x \, dx$, then the value of k is
(A) 4
(B) 2
(C) 1
(D) 0
$\int \frac{dx}{\sin^2 2x \cdot \cos^2 2x}$ equals : (a) $\frac{1}{2} [\tan 2x + \cot 2x] + C$ (b) $\tan 2x - \cot 2x + C$ (c) $\frac{1}{2} [\tan 2x - \cot 2x] + C$ (d) $\frac{1}{2} [\cot 2x - \tan 2x] + C$
$\int_{-\pi/4}^{\pi/4} \sin^3 x \, dx$ equals : (a) $2 \int_{0}^{\pi/4} \sin^3 x \, dx$ (b) 0 (c) 1 (d) $\int_{0}^{\pi/4} \sin^3 x \, dx$
Find: $\int \frac{dx}{1 + \cot x}$
Find : $\int \frac{\text{x}}{(\text{x}^{2} + 1)(\text{x} - 1)} \text{dx}$
Evaluate : $\int_{1}^{3} \frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}} dx$
Evaluate : $\int_{1}^{e} \frac{1}{\sqrt{4x^{2}-(x \log x)^{2}}} dx$
$\int \frac{2 \cos 2x - 1}{1 + 2 \sin x} dx$ is equal to :
(a) $x - 2 \cos x + C$
(b) $x + 2 \cos x + C$
(c) $-x - 2 \cos x + C$
(d) $-x + 2 \cos x + C$
$\int_{0}^{2} \sqrt{4 - x^2} \text{ dx}$ equals :
(a) $2 \log 2$
(b) $-2 \log 2$
(c) $\frac{\pi}{2}$
(d) $\pi$
Find : $\int \frac{e^{x}}{\sqrt{e^{2x} - 4e^{x} - 5}} dx$
Evaluate : $\int_{-\pi/2}^{\pi/2} \frac{\sin^{100} x}{\sin^{100} x + \cos^{100} x} dx$
$\int e^{-x} \left$ \frac{x+1}{x^2} \right$ dx$ is equal to :
(a) $\frac{e^{-x}}{x} + C$
(b) $\frac{e^x}{x} + C$
(c) $\frac{e^x}{x^2} + C$
(d) $-\frac{e^{-x}}{x} + C$
The value of $\int_{0}^{\pi/2} \log \tan x \, dx$ is :
(a) $\frac{\pi}{2}$
(b) $0$
(c) $-\frac{\pi}{2}$
(d) $1$
Find : $\int \frac{x^2 + x + 1}{(x + 1)^2 (x + 2)} \, dx$
(a) Evaluate : $\int_{\pi/4}^{\pi/2} e^{2x} \left(\frac{1 - \sin 2x}{1 - \cos 2x}\right) \, dx$
OR
(b) Evaluate : $\int_{-2}^{2} \frac{x^2}{1 + 5^x} \, dx$
(a) Find : $\int \frac{e^x}{\sqrt{5 - 4e^x - e^{2x}}} \, dx$
OR
(b) Evaluate : $\int_{0}^{\pi/2} \sqrt{\sin x} \cos^5 x \, dx$
$\int e^{5 \log x} dx$ is equal to :
(a) $\frac{x^5}{5} + C$
(b) $\frac{x^6}{6} + C$
(c) $5x^4 + C$
(d) $6x^5 + C$
$\int_{0}^{4} (e^{2x} + x) dx$ is equal to :
(a) $\frac{15 + e^8}{2}$
(b) $\frac{16 - e^8}{2}$
(c) $\frac{e^8 - 15}{2}$
(d) $\frac{-e^8 - 15}{2}$
Find :
$$\int \frac{x^2}{x^2 + 6x + 12} dx$$
Find :
$$\int \frac{2}{(1-x)(1+x^2)} \, dx$$
(a) Evaluate :
$$\int_{1/3}^{1} \frac{(x - x^3)^{1/3}}{x^4} \, dx$$
OR
(b) Evaluate :
$$\int_{1}^{3} \{|x - 1| + |x - 2|\} \, dx$$
(a) Evaluate :
$$\int_{\pi/4}^{\pi/2} e^{2x} \left$ \frac{1 - \sin 2x}{1 - \cos 2x} \right$ dx$$
OR
(b) Evaluate :
$$\int_{-2}^{2} \frac{x^2}{1 + 5^x} dx$$
$\int \frac{\sec x}{\sec x - \tan x} dx$ equals
(a) $\sec x - \tan x + c$
(b) $\sec x + \tan x + c$
(c) $\tan x - \sec x + c$
(d) $-(\sec x + \tan x) + c$
$\int_{-1}^{1} \frac{|x - 2|}{x - 2} dx, x \neq 2$ is equal to
(a) $1$
(b) $-1$
(c) $2$
(d) $-2$
Assertion (A) : $\int_{2}^{8} \frac{\sqrt{10-x}}{\sqrt{x} + \sqrt{10-x}} dx = 3$
Reason (R) : $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$
(a) Evaluate : $\int_{0}^{2\pi} \frac{1}{1+e^{\sin x}} dx$ OR (b) Find : $\int \frac{x^{4}}{(x-1)(x^{2}+1)} dx$
Evaluate : $\int\limits_0^{\frac{\pi}{2}} \sin 2x \tan^{-1}(\sin x)\,dx$
$\int \frac{\sec x}{\sec x - \tan x} \, dx$ equals
(a) $\sec x - \tan x + c$
(b) $\sec x + \tan x + c$
(c) $\tan x - \sec x + c$
(d) $-(\sec x + \tan x) + c$
The value of $\int_{0}^{\frac{\pi}{4}} (\sin 2x) \, dx$ is
(a) $0$
(b) $1$
(c) $\frac{1}{2}$
(d) $-\frac{1}{2}$
Assertion – Reason Based Questions
In the following questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
(b) Both (A) and (R) are t...
(a) Evaluate : $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\cos 2x}{1 + \cos 2x} \, dx$
OR
(b) Find : $\int e^{x^2} (x^5 + 2x^3) \, dx$
$\int \frac{\cos 2x}{\sin^2 x \cdot \cos^2 x} \, \text{d}x$ is equal to
(a) $\tan x - \cot x + C$
(b) $-\cot x - \tan x + C$
(c) $\cot x + \tan x + C$
(d) $\tan x - \cot x - C$
$\int \frac{\cos 2x}{\sin^2 x \cdot \cos^2 x} \, dx$ is equal to
(a) $\tan x - \cot x + C$
(b) $-\cot x - \tan x + C$
(c) $\cot x + \tan x + C$
(d) $\tan x - \cot x - C$
Find : $\int \frac{e^x}{\sqrt{5 - 4e^x - e^{2x}}} dx$
$\int \frac{\cos 2x}{\sin^2 x \cdot \cos^2 x} \text{dx}$ is equal to
(a) $\tan x - \cot x + C$
(b) $-\cot x - \tan x + C$
(c) $\cot x + \tan x + C$
(d) $\tan x - \cot x - C$
Evaluate : $\int_{0}^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \text{ dx}$
Evaluate : $\int_{1}^{3} (|x - 1| + |x - 2|) \text{ dx}$
Find : $\int \frac{x}{(x^2 + 1)(x - 1)} \text{ dx}$
$\int \cot^{2} x \, dx$ equals: (a) $-\cot x + x + C$ (b) $-\cot x - x + C$ (c) $\cot x - x + C$ (d) $\cot x + x + C$
The value of $\int_{-1}^{1} \frac{dx}{5 + 2x + x^{2}}$ is: (a) $\frac{\pi}{8}$ (b) $\frac{\pi}{4}$ (c) $\frac{\pi}{6}$ (d) $\frac{\pi}{2}$
Find : $\int \frac{x - 5}{(x - 2)^4} e^x \, dx$
(a) Find : $\int \frac{2}{(1 - x)(1 + x^2)} dx$
OR
(b) Find : $\int \sqrt{3 - 2x - x^2} dx$
Evaluate : $\int_{0}^{\pi} \frac{x \tan x}{\sec x + \tan x} dx$
(a) Find : $\int \frac{\text{d}x}{\sqrt{x} + \sqrt[3]{x}}$.
OR
(b) Evaluate : $\int_{0}^{\pi/2} \frac{\cos x}{(1 + \sin x)(4 + \sin x)}\text{d}x$.
Evaluate : $\int_0^\pi \frac{x}{1 + \sin x} \, \text{d}x$.
(a) Find : $\int \frac{x^3 + x}{x^4 - 9} dx$. OR (b) Evaluate, using properties : $\int_{-\pi}^{\pi} (3 \sin x - 2)^2 dx$.
(a) If $\frac{d}{dx}[F(x)] = \frac{\sec^4 x}{\csc^4 x}$ and $F$\frac{\pi}{4}$ = \frac{\pi}{4}$, then find $F(x)$.
OR
(b) Find: $\int \frac{\log x - 3}{(\log x)^4} dx$.
(a) Find: $\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$. OR (b) Evaluate: $\int_{0}^{\pi/2} \frac{\cos x}{(1 + \sin x) (4 + \sin x)} dx$.
Find : $\int \frac{\sin 3x}{\sin x} \text{dx}$
OR Evaluate : $\int_{0}^{\frac{1}{2} \log 3} \frac{e^{x}}{e^{2x} + 1} \text{dx}$
(a) Find : $\int \frac{\sin 3x}{\sin x} dx$ OR (b) Evaluate : $\int_{0}^{\frac{1}{2}\log 3} \frac{e^x}{e^{2x} + 1} dx$
Evaluate: $\int_{0}^{\pi/2} \frac{x}{\sin x + \cos x} dx$
The value of $\int_{2}^{3} \frac{x}{x^2+1}dx$ is
(a) $\log4$ (b) $\log\frac{3}{2}$ (c) $\frac{1}{2}\log2$ (d) $\log\frac{9}{4}$
Find: $\int \frac{dx}{\sqrt{3 - 2x - x^2}}$
Evaluate: $\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}$
OR
Evaluate: $\int_{0}^{4} |x - 1| \, dx$
Find $\int \frac{(x^3 + x + 1)}{(x^2 - 1)} \, dx$
(a) Find : $\int \frac{\sqrt{\cot x}}{\sin x \cos x} \text{dx}$
(b) Find : $\int \frac{1}{x(x^2 + 4)} \text{dx}$
(a) Evaluate : $\int_{0}^{1} \tan^{-1} x \text{dx}$
(b) Find : $\int \frac{2x}{x^2 + 3x + 2} \text{dx}$
Evaluate : $\int_{0}^{2\pi} \frac{dx}{1 + e^{\sin x}}$
Evaluate : $\int_{-1}^{2} |x^3 - x| dx$
Find : $\int \frac{ ext{dx}}{x^2 - 6x + 13}$
Evaluate : $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin | x | + \cos | x |) \,\text{d}x$
Find : $\int \frac{x^2}{(x^2 + 1)(3x^2 + 4)} \,\text{d}x$
OR
Evaluate : $\int_{-2}^{1} \sqrt{5 - 4x - x^2} \,\text{d}x$
Evaluate : $\int\limits_{0}^{1} x^2 e^x \, dx$
(a) Find : $\int \frac{1}{e^x + 1} \, dx$ \
OR \
(b) Evaluate : $\int\limits_{1}^{4} \left( |x| + |3 - x| \right) dx$
Evaluate: $\int_{0}^{\pi/2} (2 \log \cos x - \log \sin 2x) dx$
(a) Find : $\int \frac{1}{e^x + 1} dx$
OR
(b) Evaluate : $\int_{1}^{4} \{|x| + |3 - x|\} dx$
Evaluate : $\int_{0}^{\pi} \frac{x}{9 \sin^2 x + 16 \cos^2 x} dx$
(a) Find : $\int \frac{1}{e^x + 1} \, dx$
OR
(b) Evaluate : $\int_{1}^{4} (|x| + |3 - x|) \, dx$
Evaluate : $\int_{-3}^{3} \frac{x^4}{1 + e^x} dx$
Find : $\int \frac{dx}{\sqrt{4x-x^2}}$
Evaluate: $\int_{0}^{\pi/2} \frac{1}{1 + \cot^{5/2} x} dx$
Find : ∫ sin x / sin(x - 2a) dx
Evaluate : $\int_{0}^{5} x \cdot \sqrt{5 - x} dx$
Find :
$\int \sin^{-1} x \, dx$
(a) Find : $\int \cos x \cdot \tan^{-1}(\sin x) dx$ OR (b) Find : $\int \frac{e^x}{(e^x + 1)(e^x + 3)} dx$
Evaluate: $\int_{1}^{2} \log \left$ \frac{3}{x} - 1 \right$ \, dx$
Find : $\int \sin 2x \sin 3x \, dx$
(a) Find : $\int \frac{x^3}{\sqrt{1 - x^8}} dx$
OR (b) Find : $\int \frac{e^x \cdot x}{(x + 1)^2} dx$
Evaluate : $\int_{-4}^{4} |x + 2| \text{d}x$
Find : $\int \frac{x}{(x + 2)(3 - 2x)} dx$