Integrals Class 12 Worksheet with Answers & Notes

Integration as inverse process of differentiation

Subtopic

Integration as inverse process of differentiation under Integrals for Grade 12 CBSE.

Preview questions (no answers)

1.
$8x^7$
A. $x^8 + C$
B. $56x^6 + C$
C. $8x^8 + C$
D. $\frac{x^8}{8} + C$
2.
$\frac{1}{1+x^2}$
A. $\sin^{-1} x + C$
B. $\log(1+x^2) + C$
C. $\tan^{-1} x + C$
D. $\cot^{-1} x + C$
3.
$\frac{1}{\sqrt{1-x^2}}$
A. $\cos^{-1} x + C$
B. $\sin^{-1} x + C$
C. $\tan^{-1} x + C$
D. $\sec^{-1} x + C$
4.
$x^{1/4}$
A. $\frac{4}{5} x^{5/4} + C$
B. $\frac{1}{4} x^{-3/4} + C$
C. $\frac{5}{4} x^{5/4} + C$
D. $x^{5/4} + C$
5.
$\int \frac{x^3 + 5x^2 - 4}{x^2} dx$
A. $\frac{x^2}{2} + 5x + \frac{4}{x} + C$
B. $\frac{x^2}{2} + 5x - \frac{4}{x} + C$
C. $x^2 + 5x + \frac{4}{x} + C$
D. $\frac{x^2}{2} - 5x + \frac{4}{x} + C$
6.
$\int \frac{dx}{\cos^2 x \sin^2 x}$
A. $\tan x - \cot x + C$
B. $-(\tan x + \cot x) + C$
C. $\cot x - \tan x + C$
D. $\sec x \csc x + C$
7.
$\int \frac{x^2 - 1}{x^2 + 1} dx$
A. $x - 2\tan^{-1} x + C$
B. $x + 2\tan^{-1} x + C$
C. $x - \tan^{-1} x + C$
D. $2\tan^{-1} x - x + C$
8.
$\int \frac{dx}{x(x^4-1)}$
A. $\frac{1}{4} \ln|\frac{x^4-1}{x^4}| + C$
B. $\frac{1}{4} \ln|\frac{x^4}{x^4-1}| + C$
C. $\ln|x^4-1| - \ln|x^4| + C$
D. $\frac{1}{2} \tan^{-1}(x^2) + C$
9.
$\int \frac{x^2+1}{x^4+1} dx$
A. $\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x^2-1}{\sqrt{2}x}\right) + C$
B. $\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x-1/x}{\sqrt{2}}\right) + C$
C. $\frac{1}{2\sqrt{2}} \ln\left|\frac{x^2-\sqrt{2}x+1}{x^2+\sqrt{2}x+1}\right| + C$
D. $\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x+1/x}{\sqrt{2}}\right) + C$
10.
$f'(x) = \frac{1}{x+x^5}$
A. $\ln|x| - \frac{1}{4}\ln(1+x^4) + \frac{1}{4}\ln 2$
B. $\ln|x| - \frac{1}{4}\ln(1+x^4) - \frac{1}{4}\ln 2$
C. $\ln|x| + \ln(1+x^4) - \ln 2$
D. $\frac{1}{4}\ln|\frac{x^4}{1+x^4}| + \frac{1}{4}\ln 2$

Download the worksheet for Integrals - Integration as inverse process of differentiation to practice offline. It includes additional chapter-level practice questions.

Integration of a variety of functions by substitution, by partial fractions and by parts

Subtopic

Integration of a variety of functions by substitution, by partial fractions and by parts under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int \frac{dx}{x^2+9}$
A. $\frac{1}{3} \tan^{-1}(\frac{x}{3}) + C$
B. $\tan^{-1}(\frac{x}{3}) + C$
C. $\frac{1}{9} \tan^{-1}(\frac{x}{9}) + C$
D. $\frac{1}{3} \sin^{-1}(\frac{x}{3}) + C$
2.
$\int \frac{x+2}{x+1} dx$
A. $x + \ln|x+1| + C$
B. $x - \ln|x+1| + C$
C. $\ln|x+1| + C$
D. $x + 2\ln|x+1| + C$
3.
$\int \frac{dx}{\sqrt{9-x^2}}$
A. $\sin^{-1}(\frac{x}{3}) + C$
B. $\frac{1}{3}\sin^{-1}(\frac{x}{3}) + C$
C. $\sin^{-1}(3x) + C$
D. $\ln|x + \sqrt{9-x^2}| + C$
4.
$\int \sec^2 x \tan x dx$
A. $\frac{\tan^2 x}{2} + C$
B. $\frac{\sec^2 x}{2} + C$
C. $\tan^2 x + C$
D. $\ln|\sec x| + C$
5.
$f(x) = \frac{\cos x}{1 + \sin^2 x}$
A. $\tan^{-1}(\sin x) + C$
B. $\sin^{-1}(\cos x) + C$
C. $\ln|1 + \sin^2 x| + C$
D. $\frac{1}{2} \tan^{-1}(\sin x) + C$
6.
$\int \frac{\cos x - \sin x}{\cos x + \sin x} dx$
A. $\ln|\cos x + \sin x| + C$
B. $\ln|\cos x - \sin x| + C$
C. $-\ln|\cos x + \sin x| + C$
D. $\sin x + \cos x + C$
7.
$\int \sqrt{x^2-a^2} dx$
A. $\frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}| + C$
B. $\frac{x}{2}\sqrt{x^2-a^2} + \frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}| + C$
C. $\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}(\frac{x}{a}) + C$
D. $\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln|x+\sqrt{x^2+a^2}| + C$
8.
$\int \frac{\ln x}{x^2} dx$
A. $\frac{1}{x}(\ln x + 1) + C$
B. $-\frac{1}{x}(\ln x + 1) + C$
C. $-\frac{\ln x}{x} + x + C$
D. $\frac{\ln x}{x} - \frac{1}{x} + C$
9.
$f(x) = \frac{1}{x^2+4x+13}$
A. $\frac{1}{3} \tan^{-1}(\frac{x+2}{3}) + C$
B. $\frac{1}{2} \tan^{-1}(\frac{x+2}{2}) + C$
C. $\tan^{-1}(\frac{x+2}{3}) + C$
D. $\frac{1}{3} \ln|x^2+4x+13| + C$
10.
$\int \sqrt{x^2+2x+5} dx$
A. $\frac{x+1}{2}\sqrt{x^2+2x+5} + 2\ln|x+1+\sqrt{x^2+2x+5}| + C$
B. $\frac{x+1}{2}\sqrt{x^2+2x+5} + 4\ln|x+1+\sqrt{x^2+2x+5}| + C$
C. $(x+1)\sqrt{x^2+2x+5} + 2\ln|x+1+\sqrt{x^2+2x+5}| + C$
D. $\frac{x+1}{2}\sqrt{x^2+2x+5} - 2\ln|x+1+\sqrt{x^2+2x+5}| + C$

Download the worksheet for Integrals - Integration of a variety of functions by substitution, by partial fractions and by parts to practice offline. It includes additional chapter-level practice questions.

Evaluation of simple integrals and problems based on them

Subtopic

Evaluation of simple integrals and problems based on them under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int (e^x - \sin x) dx$
A. $e^x + \cos x + C$
B. $e^x - \cos x + C$
C. $e^x + \sin x + C$
D. $e^x - \sin x + C$
2.
$\int \frac{1}{3x} dx$
A. $\frac{1}{3} \ln |x| + C$
B. $3 \ln |x| + C$
C. $\ln |3x| + C$
D. $\frac{1}{3x^2} + C$
3.
$\int (x^3 - 8) dx$
A. $\frac{x^4}{4} - 8x + C$
B. $3x^2 + C$
C. $\frac{x^4}{4} + C$
D. $x^4 - 8x + C$
4.
$\int \sin(x/4) dx$
A. $-4 \cos(x/4) + C$
B. $4 \cos(x/4) + C$
C. $-\frac{1}{4} \cos(x/4) + C$
D. $\frac{1}{4} \cos(x/4) + C$
5.
$\int \frac{\cot x}{\log(\sin x)} dx$
A. $\log|\log(\sin x)| + C$
B. $\log|\cot x| + C$
C. $(\log(\sin x))^2 + C$
D. $\csc^2 x + C$
6.
$\int \frac{dx}{\cos(x-a)\cos(x-b)}$
A. $\frac{1}{\sin(a-b)} \log|\frac{\cos(x-a)}{\cos(x-b)}| + C$
B. $\frac{1}{\cos(a-b)} \log|\frac{\sin(x-a)}{\sin(x-b)}| + C$
C. $\frac{1}{\sin(a-b)} \log|\frac{\cos(x-b)}{\cos(x-a)}| + C$
D. $\log|\sec(x-a)\sec(x-b)| + C$
7.
$\int \frac{e^x(1+x)}{\cos^2(xe^x)} dx$
A. $\tan(xe^x) + C$
B. $\cot(xe^x) + C$
C. $\sec(xe^x) + C$
D. $\tan(e^x) + C$
8.
$\int \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$
A. $\sin^{-1}(\sin x - \cos x) + C$
B. $\sin^{-1}(\cos x - \sin x) + C$
C. $\cos^{-1}(\sin x - \cos x) + C$
D. $\ln |\sin x - \cos x + \sqrt{\sin 2x}| + C$
9.
$\int_0^{1} \cot^{-1}(1-x+x^2) dx$
A. $\frac{\pi}{2} - \ln 2$
B. $\frac{\pi}{4} - \ln 2$
C. $\pi - \ln 2$
D. $\frac{\pi}{2} + \ln 2$
10.
$\int \frac{\ln x}{(1+\ln x)^2} dx$
A. $\frac{x}{1+\ln x} + C$
B. $\frac{1}{1+\ln x} + C$
C. $\frac{\ln x}{1+\ln x} + C$
D. $x(1+\ln x) + C$

Download the worksheet for Integrals - Evaluation of simple integrals and problems based on them to practice offline. It includes additional chapter-level practice questions.

Definite integrals as a limit of a sum

Subtopic

Definite integrals as a limit of a sum under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} (\frac{r}{n})^8$
A. $\frac{1}{8}$
B. $\frac{1}{9}$
C. $\frac{1}{10}$
D. $\frac{1}{7}$
2.
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} \cos\left(\frac{r\pi}{n}\right)$
A. $1$
B. $0$
C. $\pi$
D. $\frac{1}{\pi}$
3.
$h = 0.1/n$
A. $1$
B. $0.1$
C. $10$
D. $n$
4.
$f(x)$
A. $x^{10}$
B. $10x$
C. $x^{11}$
D. $10^x$
5.
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n \frac{1}{1 + \sqrt{r/n}}$
A. $2(1 - \ln 2)$
B. $1 - \ln 2$
C. $2(1 + \ln 2)$
D. $2\ln 2 - 2$
6.
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n \frac{1}{\sqrt{3 + r/n}}$
A. $4 - 2\sqrt{3}$
B. $2 - \sqrt{3}$
C. $4 + 2\sqrt{3}$
D. $2\sqrt{3} - 4$
7.
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n \frac{1}{(1+r/n)^3}$
A. $\frac{1}{2}$
B. $\frac{3}{8}$
C. $\frac{1}{4}$
D. $\frac{5}{8}$
8.
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} \sqrt{\frac{n-r}{n+r}}$
A. $\frac{\pi}{2} - 1$
B. $\frac{\pi}{2} + 1$
C. $\pi - 2$
D. $\frac{\pi}{4}$
9.
$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} e^{r/n} \sin\left( \frac{r}{n} \right)$
A. $\frac{e(\sin 1 - \cos 1) + 1}{2}$
B. $\frac{e(\sin 1 + \cos 1) - 1}{2}$
C. $e \sin 1$
D. $\frac{e \cos 1 + 1}{2}$
10.
$\lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n^2 + nr + r^2}$
A. $\frac{\pi}{3\sqrt{3}}$
B. $\frac{\pi}{\sqrt{3}}$
C. $\frac{2\pi}{3\sqrt{3}}$
D. $\frac{\pi}{6}$

Download the worksheet for Integrals - Definite integrals as a limit of a sum to practice offline. It includes additional chapter-level practice questions.

Fundamental Theorem of Calculus

Subtopic

Fundamental Theorem of Calculus under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int_1^2 \frac{3}{x^4} \, dx$
A. $\frac{7}{8}$
B. $\frac{15}{16}$
C. $\frac{1}{2}$
D. $\frac{3}{4}$
2.
$\int_0^{\pi/2} \sin x \, dx$
A. $0$
B. $1$
C. $-1$
D. $\pi$
3.
$\int_0^1 (x^2 + x + 1) \, dx$
A. $\frac{11}{6}$
B. $\frac{5}{6}$
C. $\frac{7}{6}$
D. $2$
4.
$\int_{-1}^0 e^{-x} \, dx$
A. $1-e$
B. $e-1$
C. $e$
D. $1$
5.
$\int_0^{\sqrt{3}} \frac{1}{\sqrt{4 - x^2}} dx$
A. $\frac{\pi}{6}$
B. $\frac{\pi}{3}$
C. $\frac{\pi}{4}$
D. $\frac{\pi}{2}$
6.
$f(x) = \int_x^{x^2} \ln t dt$
A. $(4x-1)\ln x$
B. $(2x-1)\ln x$
C. $4x \ln x$
D. $2x \ln(x^2) - \ln x$
7.
$\int_0^{\frac{\pi}{2}} \frac{\sin x - \cos x}{1 + \sin x \cos x} dx$
A. $1$
B. $\frac{\pi}{2}$
C. $0$
D. $-1$
8.
$\int_{0}^{x^2(1+x)} f(t) dt = x$
A. $1/5$
B. $1/2$
C. $2/5$
D. $1/4$
9.
$\frac{d}{dx} \left[ \int_{x}^{x^2} t \ln t dt \right]$
A. $2x^3 \ln x^2 - x \ln x$
B. $4x^3 \ln x - x \ln x$
C. $x^2 \ln x^2 - x \ln x$
D. $4x^3 \ln x + x \ln x$
10.
$F(x) = \int_{0}^{x} f(t) dt$
A. $\frac{1}{2}(\cos x + \sin x - e^{-x})$
B. $\cos x - \sin x$
C. $e^{-x} \cos x$
D. $\frac{1}{2}(\cos x - \sin x + e^{-x})$

Download the worksheet for Integrals - Fundamental Theorem of Calculus to practice offline. It includes additional chapter-level practice questions.

Basic properties of definite integrals and evaluation of definite integrals

Subtopic

Basic properties of definite integrals and evaluation of definite integrals under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int_{a}^{b} f(x) dx$
A. $\int_{a}^{b} f(a+b-x) dx$
B. $\int_{a}^{b} f(x-a-b) dx$
C. $\int_{a}^{b} f(a-x) dx$
D. $\int_{a}^{b} f(b-x) dx$
2.
$\int_{0}^{1} (x + 1)^2 dx$
A. $\frac{7}{3}$
B. $\frac{8}{3}$
C. $1$
D. $2$
3.
$\int_{1}^{4} \frac{1}{\sqrt{x}} dx$
A. $1$
B. $2$
C. $3$
D. $4$
4.
$\int_{0}^{2} e^{x/2} dx$
A. $e - 1$
B. $2(e - 1)$
C. $2e$
D. $e$
5.
$\int_{0}^{2} \frac{dx}{x^2 + 4}$
A. $\frac{\pi}{2}$
B. $\frac{\pi}{4}$
C. $\pi$
D. $\frac{\pi}{8}$
6.
$\int_{0}^{\pi/4} \log(1 + \tan x) \, dx$
A. $\frac{\pi}{4} \log 2$
B. $\frac{\pi}{8} \log 2$
C. $\frac{\pi}{2} \log 2$
D. $\pi \log 2$
7.
$\int_{1}^{2} e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) \, dx$
A. $\frac{e^2}{2} - e$
B. $e^2 - e$
C. $\frac{e^2 - e}{2}$
D. $e^2$
8.
$\int_{0}^{1} \frac{dx}{(1+x^2)^2}$
A. $\frac{\pi}{8} + \frac{1}{4}$
B. $\frac{\pi}{4} + \frac{1}{2}$
C. $\frac{\pi}{8} - \frac{1}{4}$
D. $\frac{\pi}{16}$
9.
$\int_{0}^{\pi/2} \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}$
A. $\frac{\pi}{ab}$
B. $\frac{\pi}{2ab}$
C. $\frac{\pi}{4ab}$
D. $\frac{2\pi}{ab}$
10.
$\int_{0}^{1} \frac{1-x}{1+x} \frac{dx}{\sqrt{x}}$
A. $\pi - 2$
B. $\pi + 2$
C. $\frac{\pi}{2} - 1$
D. $2 - \pi$

Download the worksheet for Integrals - Basic properties of definite integrals and evaluation of definite integrals to practice offline. It includes additional chapter-level practice questions.

Some properties of indefinite integral

Subtopic

Some properties of indefinite integral under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int (2x + \sin x) \, dx$
A. $Power rule$
B. $Substitution rule$
C. $Linearity property$
D. $Integration by parts$
2.
$\frac{d}{dx} \int \pi \, dx$
A. $0$
B. $\pi$
C. $\pi x$
D. $\pi + C$
3.
$\int \frac{d}{dx} (x^3 + 3x^2 + 3x + 1) \, dx$
A. $3x^2 + 6x + 3 + C$
B. $x^3 + 3x^2 + 3x + 1 + C$
C. $(x+1)^3 + C$
D. $Both B and C are correct$
4.
$\int f(x) \, dx = \Psi(x) + C$
A. $7 \Psi(x) + C$
B. $\Psi(7x) + C$
C. $7 \Psi(x) + 7C$
D. $Both A and C are acceptable forms$
5.
$\int f(x) dx = g(x) + C$
A. $\frac{1}{2} g(2x) + C$
B. $g(2x) + C$
C. $2g(2x) + C$
D. $\frac{1}{2} g(x) + C$
6.
$\int (3^x \cdot e^x) dx$
A. $\frac{(3e)^x}{\ln 3 + 1} + C$
B. $\frac{3^x e^x}{\ln 3} + C$
C. $(3e)^x + C$
D. $\frac{(3e)^x}{\ln 3} + C$
7.
$\int \frac{x^4}{x^2+1} dx$
A. $\frac{x^3}{3} - x + \tan^{-1} x + C$
B. $\frac{x^3}{3} + x - \tan^{-1} x + C$
C. $\frac{x^3}{3} - x - \tan^{-1} x + C$
D. $x^3 - x + \tan^{-1} x + C$
8.
$\int \frac{dx}{\sin(x-a)\sin(x-b)} = \frac{1}{\sin(a-b)} \ln |f(x)| + C$
A. $\frac{\sin(x-a)}{\sin(x-b)}$
B. $\frac{\sin(x-b)}{\sin(x-a)}$
C. $\frac{\cos(x-a)}{\cos(x-b)}$
D. $\frac{\tan(x-a)}{\tan(x-b)}$
9.
$\int \frac{\cos x - \sin x}{1+\sin 2x} dx$
A. $\frac{-1}{\sin x + \cos x} + C$
B. $\frac{1}{\sin x + \cos x} + C$
C. $\ln|\sin x + \cos x| + C$
D. $\sin x + \cos x + C$
10.
$\int \frac{\ln(x+\sqrt{x^2+1})}{\sqrt{x^2+1}} dx$
A. $\frac{1}{2} [\ln(x+\sqrt{x^2+1})]^2 + C$
B. $\ln(x+\sqrt{x^2+1}) + C$
C. $2\ln(x+\sqrt{x^2+1}) + C$
D. $\sqrt{x^2+1} \ln(x+\sqrt{x^2+1}) + C$

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Methods of Integration

Subtopic

Methods of Integration under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int \frac{dx}{x^2-9}$
A. $\frac{1}{3} \log|\frac{x-3}{x+3}| + C$
B. $\frac{1}{6} \log|\frac{x+3}{x-3}| + C$
C. $\frac{1}{6} \log|\frac{x-3}{x+3}| + C$
D. $\frac{1}{3} \tan^{-1}(\frac{x}{3}) + C$
2.
$\int x \cos(x^2) \, dx$
A. $\frac{1}{2} \sin(x^2) + C$
B. $\sin(x^2) + C$
C. $2 \sin(x^2) + C$
D. $-\frac{1}{2} \sin(x^2) + C$
3.
$\int \frac{1}{\sqrt{1-4x^2}} \, dx$
A. $\sin^{-1}(2x) + C$
B. $\frac{1}{2} \sin^{-1}(2x) + C$
C. $2 \sin^{-1}(2x) + C$
D. $\frac{1}{2} \tan^{-1}(2x) + C$
4.
$\int \frac{\cos x - \sin x}{\cos x + \sin x} \, dx$
A. $\log|\cos x - \sin x| + C$
B. $\log|\cos x + \sin x| + C$
C. $-\log|\cos x + \sin x| + C$
D. $\tan x + C$
5.
$\int \frac{dx}{(x+1)^2 + 1}$
A. $\tan^{-1}(x+1) + C$
B. $\log |(x+1)^2 + 1| + C$
C. $\frac{1}{x+1} + C$
D. $\sin^{-1}(x+1) + C$
6.
$\int \frac{e^x}{1 + e^{2x}} dx$
A. $\tan^{-1}(e^x) + C$
B. $\log |1 + e^{2x}| + C$
C. $e^x \tan^{-1}(e^x) + C$
D. $\frac{1}{2} \tan^{-1}(e^x) + C$
7.
$\int \frac{dx}{x^2 + a^2}$
A. $\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$
B. $\tan^{-1}(\frac{x}{a}) + C$
C. $\frac{1}{a} \sin^{-1}(\frac{x}{a}) + C$
D. $\frac{1}{2a} \log |\frac{x-a}{x+a}| + C$
8.
$\int e^x \frac{1-x^2}{(1+x^2)^2} dx$
A. $\frac{e^x}{1+x^2} + C$
B. $\frac{-e^x}{1+x^2} + C$
C. $\frac{xe^x}{1+x^2} + C$
D. $\frac{-xe^x}{1+x^2} + C$
9.
$\int \frac{dx}{x \sqrt{x^4-1}}$
A. $\frac{1}{2} \sec^{-1}(x^2) + C$
B. $\sec^{-1}(x^2) + C$
C. $\frac{1}{2} \tan^{-1}(x^2) + C$
D. $\frac{1}{2} \ln|x^2 + \sqrt{x^4-1}| + C$
10.
$\int \frac{dx}{a^2 \sin^2 x + b^2 \cos^2 x}$
A. $\frac{1}{ab} \tan^{-1}(\frac{a \tan x}{b}) + C$
B. $\frac{1}{a} \tan^{-1}(\frac{a \tan x}{b}) + C$
C. $\frac{a}{b} \tan^{-1}(\frac{a \tan x}{b}) + C$
D. $\frac{1}{ab} \tan^{-1}(\tan x) + C$

Download the worksheet for Integrals - Methods of Integration to practice offline. It includes additional chapter-level practice questions.

Integration using trigonometric identities

Subtopic

Integration using trigonometric identities under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int \frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} \, dx$
A. $\sec x - \csc x + C$
B. $\sec x + \csc x + C$
C. $\tan x - \cot x + C$
D. $\sin x - \cos x + C$
2.
$\int \cos 2x \cos 4x \, dx$
A. $\frac{\sin 6x}{12} + \frac{\sin 2x}{4} + C$
B. $\frac{\sin 6x}{6} + \frac{\sin 2x}{2} + C$
C. $\frac{\cos 6x}{12} + \frac{\cos 2x}{4} + C$
D. $\frac{\sin 6x}{12} - \frac{\sin 2x}{4} + C$
3.
$\int (\sin x + \cos x)^2 \, dx$
A. $x - \frac{1}{2} \cos 2x + C$
B. $x + \frac{1}{2} \cos 2x + C$
C. $x - \cos 2x + C$
D. $x + \cos 2x + C$
4.
$\int \sin 4x \sin 2x \, dx$
A. $\frac{1}{2} \int (\cos 2x - \cos 6x) \, dx$
B. $\frac{1}{2} \int (\cos 6x - \cos 2x) \, dx$
C. $\frac{1}{2} \int (\sin 6x + \sin 2x) \, dx$
D. $\frac{1}{2} \int (\sin 6x - \sin 2x) \, dx$
5.
$\int \sin^3(x) dx$
A. $\frac{1}{4} \cos(3x) - \frac{3}{4} \cos(x) + C$
B. $\frac{1}{12} \cos(3x) - \frac{3}{4} \cos(x) + C$
C. $\frac{3}{4} \cos(x) - \frac{1}{12} \cos(3x) + C$
D. $-\frac{3}{4} \cos(x) + \frac{1}{12} \cos(3x) + C$
6.
$\int \sqrt{1 + \cos(2x)} dx$
A. $\sqrt{2} \sin(x) + C$
B. $2 \cos(x) + C$
C. $\sqrt{2} \cos(x) + C$
D. $2 \sin(x) + C$
7.
$\int \frac{1}{\cos^2(x) \sin^2(x)} dx$
A. $\tan(x) + \cot(x) + C$
B. $\tan(x) - \cot(x) + C$
C. $\sec(x) + \csc(x) + C$
D. $\cot(x) - \tan(x) + C$
8.
$f(x) = \frac{\sin(2x)}{\sin^4(x) + \cos^4(x)}$
A. $\tan^{-1}(\tan^2(x)) + C$
B. $\ln|\sin^4(x) + \cos^4(x)| + C$
C. $\tan^{-1}(2\tan^2(x)) + C$
D. $\frac{1}{2} \tan^{-1}(\tan^2(x)) + C$
9.
$A(x) = \cos^6(x)$
A. $\frac{1}{16}$
B. $\frac{5}{16}$
C. $\frac{3}{8}$
D. $\frac{5}{32}$
10.
$H(x) = \frac{1}{\sin(x - a) \sin(x - b)}$
A. $\frac{1}{\sin(a-b)} \ln \left| \frac{\sin(x-a)}{\sin(x-b)} \right| + C$
B. $\frac{1}{\sin(b-a)} \ln \left| \frac{\sin(x-a)}{\sin(x-b)} \right| + C$
C. $\sin(a-b) \ln |\sin(x-a) \sin(x-b)| + C$
D. $\frac{1}{\cos(a-b)} \ln \left| \frac{\cos(x-a)}{\cos(x-b)} \right| + C$

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Integrals of Some Particular Functions

Subtopic

Integrals of Some Particular Functions under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int \frac{dx}{x\sqrt{x^2-1}}$
A. $\sin^{-1}x + C$
B. $\cos^{-1}x + C$
C. $\sec^{-1}x + C$
D. $\tan^{-1}x + C$
2.
$\frac{1}{\sqrt{x^2 + 2x + 2}}$
A. $\ln|x+1 + \sqrt{x^2+2x+2}| + C$
B. $\tan^{-1}(x+1) + C$
C. $\ln|x + \sqrt{x^2+2x+2}| + C$
D. $\sin^{-1}(x+1) + C$
3.
$\int \frac{dx}{x^2 + 4x + 5}$
A. $\tan^{-1}(x+2) + C$
B. $\ln|x^2 + 4x + 5| + C$
C. $\frac{1}{2} \tan^{-1}(\frac{x+2}{2}) + C$
D. $\sin^{-1}(x+2) + C$
4.
$\int \frac{dx}{\sqrt{1 - 4x^2}}$
A. $\sin^{-1}(2x) + C$
B. $\frac{1}{2} \sin^{-1}(2x) + C$
C. $\frac{1}{2} \ln|2x + \sqrt{1-4x^2}| + C$
D. $\cos^{-1}(2x) + C$
5.
$\int \frac{dx}{x^2 - 4x + 13}$
A. $\frac{1}{3} \tan^{-1}\left(\frac{x-2}{3}\right) + C$
B. $\frac{1}{6} \log\left|\frac{x-5}{x+1}\right| + C$
C. $\frac{1}{2} \tan^{-1}\left(\frac{x-2}{3}\right) + C$
D. $\log\left|x^2 - 4x + 13\right| + C$
6.
$\int \frac{dx}{x^2 - 6x + 13}$
A. $\frac{1}{2} \tan^{-1}(\frac{x-3}{2}) + C$
B. $\tan^{-1}(\frac{x-3}{2}) + C$
C. $\frac{1}{2} \ln|x^2 - 6x + 13| + C$
D. $\frac{1}{4} \tan^{-1}(\frac{x-3}{2}) + C$
7.
$\int \frac{dx}{\sqrt{1 + 4x - x^2}}$
A. $\sin^{-1}(\frac{x-2}{\sqrt{5}}) + C$
B. $\sin^{-1}(\frac{x+2}{\sqrt{5}}) + C$
C. $\ln|x-2 + \sqrt{1+4x-x^2}| + C$
D. $\frac{1}{\sqrt{5}} \sin^{-1}(\frac{x-2}{\sqrt{5}}) + C$
8.
$\int \frac{dx}{\sqrt{x^2 + 8x + 7}}$
A. $\ln|x+4 + \sqrt{x^2+8x+7}| + C$
B. $\sin^{-1}\left(\frac{x+4}{3}\right) + C$
C. $\ln|x-4 + \sqrt{x^2+8x+7}| + C$
D. $\frac{1}{3} \ln|x+4 + \sqrt{x^2+8x+7}| + C$
9.
$\int \frac{dx}{4x^2 + 4x + 5}$
A. $\frac{1}{4} \tan^{-1}\left(\frac{2x+1}{2}\right) + C$
B. $\frac{1}{2} \tan^{-1}\left(\frac{2x+1}{2}\right) + C$
C. $\tan^{-1}(2x+1) + C$
D. $\frac{1}{4} \ln|4x^2+4x+5| + C$
10.
$\int \frac{dx}{\sqrt{16 - 6x - x^2}}$
A. $\sin^{-1}\left(\frac{x+3}{5}\right) + C$
B. $\ln|x+3 + \sqrt{16-6x-x^2}| + C$
C. $\frac{1}{5} \sin^{-1}\left(\frac{x+3}{5}\right) + C$
D. $\sin^{-1}\left(\frac{x-3}{5}\right) + C$

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Integral of the type ∫ e^x(f(x) + f'(x)) dx

Subtopic

Integral of the type ∫ e^x(f(x) + f'(x)) dx under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int e^x (\frac{1}{2} x^2 + x) dx$
A. $e^x x^2 + C$
B. $\frac{1}{2} e^x x^2 + C$
C. $e^x x + C$
D. $e^x (\frac{1}{2}x^2 + x) + C$
2.
$\int e^x \left( \log(\csc x - \cot x) + \csc x \right) dx$
A. $e^x \csc x + C$
B. $e^x \cot x + C$
C. $e^x \log(\csc x - \cot x) + C$
D. $e^x \log(\sin x) + C$
3.
$\int e^x (\tan x + \log(\sec^2 x)) dx$
A. $2e^x \sec x + C$
B. $2e^x \tan x + C$
C. $e^x \tan^2 x + C$
D. $2e^x \sec^2 x + C$
4.
$\int e^x \left( \frac{1}{x^2+1} - \frac{2x}{(x^2+1)^2} \right) dx$
A. $\frac{e^x}{x^2+1} + C$
B. $e^x \tan^{-1} x + C$
C. $-\frac{e^x}{x^2+1} + C$
D. $\frac{e^x}{(x^2+1)^2} + C$
5.
$\int e^x (\frac{1}{x^2+1} - \frac{2x}{(x^2+1)^2}) dx$
A. $\frac{e^x}{x^2+1} + C$
B. $e^x \tan^{-1} x + C$
C. $e^x (x^2+1) + C$
D. $\frac{-e^x}{x^2+1} + C$
6.
$\int e^x (\frac{x^2-3}{(x+1)^2} + \dots)$
A. $e^x \frac{x-1}{x+1} + C$
B. $e^x \frac{x+1}{x-1} + C$
C. $\frac{e^x}{x+1} + C$
D. $e^x (x-1) + C$
7.
$\int e^x (\frac{2-\sin 2x}{1-\cos 2x}) dx$
A. $e^x \cot x + C$
B. $-e^x \cot x + C$
C. $e^x \tan x + C$
D. $-e^x \tan x + C$
8.
$\int e^x \frac{2x^2+3x+1}{(x+1)^2} dx$
A. $e^x \left(2 - \frac{1}{x+1}\right) + C$
B. $e^x \left(2 + \frac{1}{x+1}\right) + C$
C. $e^x \frac{2x+1}{x+1} + C$
D. $e^x \frac{x}{x+1} + C$
9.
$\int e^x \frac{x^3+x+2}{(x^2+1)^2} dx$
A. $\frac{e^x}{x^2+1} + C$
B. $\frac{xe^x}{x^2+1} + C$
C. $\frac{(x+1)e^x}{x^2+1} + C$
D. $\frac{e^x}{(x^2+1)^2} + C$
10.
$\int e^x \frac{x^2+7x+11}{(x+4)^2} dx$
A. $e^x \left(1 - \frac{1}{x+4}\right) + C$
B. $e^x \left(1 + \frac{1}{x+4}\right) + C$
C. $e^x \frac{x+3}{x+4} + C$
D. $e^x \frac{x+5}{x+4} + C$

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Integrals of some more types

Subtopic

Integrals of some more types under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int \sqrt{x^2 + 2x + 5} \, dx$
A. $\frac{x+1}{2}\sqrt{x^2+2x+5} + \log|x+1+\sqrt{x^2+2x+5}| + C$
B. $\frac{x+1}{2}\sqrt{x^2+2x+5} + 4 \log|x+1+\sqrt{x^2+2x+5}| + C$
C. $\frac{x+1}{2}\sqrt{x^2+2x+5} + 2 \log|x+1+\sqrt{x^2+2x+5}| + C$
D. $\frac{x+1}{2}\sqrt{x^2+2x+5} - 2 \log|x+1+\sqrt{x^2+2x+5}| + C$
2.
$\int \sqrt{5 - 4x - x^2} \, dx$
A. $\frac{x+2}{2}\sqrt{5-4x-x^2} + \frac{9}{2} \sin^{-1}\left(\frac{x+2}{3}\right) + C$
B. $\frac{x+2}{2}\sqrt{5-4x-x^2} + 9 \sin^{-1}\left(\frac{x+2}{3}\right) + C$
C. $\frac{x+2}{2}\sqrt{5-4x-x^2} + 3 \sin^{-1}\left(\frac{x+2}{3}\right) + C$
D. $\frac{x+2}{2}\sqrt{5-4x-x^2} + \frac{9}{2} \sin^{-1}\left(\frac{x+2}{9}\right) + C$
3.
$\int \sqrt{x^2 - 6x + 8} \, dx$
A. $\frac{x-3}{2}\sqrt{x^2-6x+8} + \frac{1}{2} \log|x-3+\sqrt{x^2-6x+8}| + C$
B. $\frac{x-3}{2}\sqrt{x^2-6x+8} - \log|x-3+\sqrt{x^2-6x+8}| + C$
C. $\frac{x-3}{2}\sqrt{x^2-6x+8} - \frac{1}{2} \log|x-3+\sqrt{x^2-6x+8}| + C$
D. $\frac{x-3}{2}\sqrt{x^2-6x+8} - 3 \log|x-3+\sqrt{x^2-6x+8}| + C$
4.
$\int \sqrt{x^2 + 4x + 5} \, dx$
A. $\frac{x+2}{2}\sqrt{x^2+4x+5} - \frac{1}{2} \log|x+2+\sqrt{x^2+4x+5}| + C$
B. $\frac{x+2}{2}\sqrt{x^2+4x+5} + \frac{1}{2} \log|x+2+\sqrt{x^2+4x+5}| + C$
C. $\frac{x+2}{2}\sqrt{x^2+4x+5} + \log|x+2+\sqrt{x^2+4x+5}| + C$
D. $\frac{x+2}{2}\sqrt{x^2+4x+5} + 2 \log|x+2+\sqrt{x^2+4x+5}| + C$
5.
$\int e^x \frac{1+x \log x}{x} \, dx$
A. $e^x \log x + C$
B. $e^x (1 + \log x) + C$
C. $\frac{e^x}{x} + C$
D. $e^x x \log x + C$
6.
$\int \sqrt{x^2 - 5x + 6} \, dx$
A. $\frac{2x-5}{4}\sqrt{x^2-5x+6} - \frac{1}{8}\log|x - \frac{5}{2} + \sqrt{x^2-5x+6}| + C$
B. $\frac{2x-5}{4}\sqrt{x^2-5x+6} + \frac{1}{8}\log|x - \frac{5}{2} + \sqrt{x^2-5x+6}| + C$
C. $\frac{x-5/2}{2}\sqrt{x^2-5x+6} - \frac{1}{4}\log|x - \frac{5}{2} + \sqrt{x^2-5x+6}| + C$
D. $\frac{2x-5}{4}\sqrt{x^2-5x+6} - \frac{1}{2}\log|x - \frac{5}{2} + \sqrt{x^2-5x+6}| + C$
7.
$\int \sqrt{5 - 4x - x^2} \, dx$
A. $\frac{x+2}{2}\sqrt{5-4x-x^2} + \frac{9}{2}\sin^{-1}\left(\frac{x+2}{3}\right) + C$
B. $\frac{x+2}{2}\sqrt{5-4x-x^2} + 9\sin^{-1}\left(\frac{x+2}{3}\right) + C$
C. $\frac{x-2}{2}\sqrt{5-4x-x^2} + \frac{9}{2}\sin^{-1}\left(\frac{x-2}{3}\right) + C$
D. $\frac{x+2}{2}\sqrt{5-4x-x^2} + \frac{9}{2}\log|x+2 + \sqrt{5-4x-x^2}| + C$
8.
$\int_0^1 \sqrt{\frac{1-x}{1+x}} \, dx$
A. $\frac{\pi}{2} - 1$
B. $\pi - 2$
C. $\frac{\pi}{4} - \frac{1}{2}$
D. $\frac{\pi}{2} + 1$
9.
$\int \frac{dx}{(x+1)\sqrt{1+x-x^2}}$
A. $-\sin^{-1}\left(\frac{1-x}{(x+1)\sqrt{5}}\right) + C$
B. $-\sin^{-1}\left(\frac{3x+1}{(x+1)\sqrt{5}}\right) + C$
C. $\sin^{-1}\left(\frac{x-1}{(x+1)\sqrt{5}}\right) + C$
D. $None of these$
10.
$\int \sqrt{1+2\tan x(\tan x + \sec x)} \, dx$
A. $\log|\sec x + \tan x| + \log|\sec x| + C$
B. $\log|\sec x(\sec x + \tan x)| + C$
C. $\sec x + \tan x + C$
D. $Both A and B are correct$

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Definite Integral

Subtopic

Definite Integral under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\int_{0}^{\pi/2} \cos^{2} x \, dx$
A. $\frac{\pi}{4}$
B. $\frac{\pi}{2}$
C. $1$
D. $0$
2.
$\int_{0}^{2} (x-1)^{2} \, dx$
A. $\frac{1}{3}$
B. $\frac{2}{3}$
C. $1$
D. $2$
3.
$\int_{0}^{1} \frac{1}{\sqrt{1-x^{2}}} \, dx$
A. $\pi$
B. $\frac{\pi}{2}$
C. $\frac{\pi}{4}$
D. $1$
4.
$\int_{0}^{\pi} \cos x \, dx$
A. $0$
B. $2$
C. $-2$
D. $1$
5.
$\int_{-1}^{1} \sin^5 x \cos^4 x dx$
A. $\frac{2}{9}$
B. $1$
C. $0$
D. $\frac{\pi}{2}$
6.
$\int_{0}^{\pi/2} \frac{1}{a^2 \cos^2 x + b^2 \sin^2 x} dx$
A. $\frac{\pi}{ab}$
B. $\frac{\pi}{2ab}$
C. $\frac{\pi}{2a}$
D. $\frac{\pi}{2b}$
7.
$\int_{0}^{1} x e^{-x} dx$
A. $1 - \frac{2}{e}$
B. $\frac{1}{e}$
C. $1 - \frac{1}{e}$
D. $2e$
8.
$\int_{0}^{\pi/2} \frac{\cos x}{(1+\sin x)(2+\sin x)} dx$
A. $\ln(2/3)$
B. $\ln(4/3)$
C. $\ln(3/2)$
D. $\ln(3/4)$
9.
$\int_{0}^{\pi} \frac{x \sin x}{1 + \sin x} dx$
A. $\frac{\pi^2}{2} - \pi$
B. $\pi^2 - 2\pi$
C. $\frac{\pi^2}{4} - \frac{\pi}{2}$
D. $\frac{\pi^2}{2}$
10.
$\int_{0}^{100} \{x\} dx$
A. $50$
B. $100$
C. $25$
D. $0$

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Area function

Subtopic

Area function under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$A(x) = \int_{1}^{x} \frac{1}{t} dt$
A. $0$
B. $e$
C. $1$
D. $\ln(e) - 1$
2.
$A(x) = \int_{0}^{x} \cos(t^2) dt$
A. $\sin(x^2)$
B. $2x \cos(x^2)$
C. $\cos(x^2)$
D. $-\sin(x^2)$
3.
$A(x) = \int_{-1}^{x} 2t dt$
A. $1$
B. $0$
C. $-1$
D. $2$
4.
$x$
A. $x = 0$
B. $x = 4$
C. $x = 2$
D. $x = -2$
5.
$A(x) = \int_{0}^{x} \sin(t) \cos(t) dt$
A. $1$
B. $0.5$
C. $0.25$
D. $2$
6.
$A(x) = \int_{1}^{x} \frac{1}{t^2} dt$
A. $0$
B. $1$
C. $\infty$
D. $-1$
7.
$A(x) = \int_{0}^{x} (2 - t) dt$
A. $x = 2$
B. $x = 4$
C. $x = 1$
D. $x = 3$
8.
$a(t) = 6t$
A. $3$
B. $5$
C. $12.5$
D. $25$
9.
$A(x) = \int_{0}^{x} |t| dt$
A. $A(-2) = -2, A(2) = 2$
B. $A(-2) = 2, A(2) = 2$
C. $A(-2) = -2, A(2) = -2$
D. $A(-2) = 0, A(2) = 2$
10.
$E(t) = 2t + 1$
A. $2$
B. $3$
C. $4$
D. $6$

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First fundamental theorem of integral calculus

Subtopic

First fundamental theorem of integral calculus under Integrals for Grade 12 CBSE.

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Preview questions (no answers)

1.
$\frac{d}{dx} \int_{0}^{x} \pi^{t} dt$
A. $\pi^{x} \ln \pi$
B. $\pi^{x}$
C. $\frac{\pi^{x}}{\ln \pi}$
D. $\pi^{t}$
2.
$\frac{d}{dx} \int_{5}^{x} \sqrt{t^{2}-16} dt$
A. $\sqrt{x^{2}-16}$
B. $\frac{x}{\sqrt{x^{2}-16}}$
C. $\sqrt{t^{2}-16}$
D. $3$
3.
$A(x) = \int_{0}^{x} (t + \sin t) dt$
A. $1 + \cos x$
B. $\frac{x^{2}}{2} - \cos x$
C. $x + \sin x + 1$
D. $x + \sin x$
4.
$\frac{d}{dx} \int_{0}^{x} \cos^{2}(t) dt$
A. $-2\cos(x)\sin(x)$
B. $\frac{1+\cos(2x)}{2}$
C. $\cos^{2}(x)$
D. $\cos^{2}(t)$
5.
$d/dx \int_{1}^{x^3} e^t \, dt$
A. $0$
B. $1$
C. $e$
D. $-1$
6.
$f(x) = \int_{0}^{x} \arctan(t) \, dt$
A. $\pi/4$
B. $\pi/2$
C. $1$
D. $0$
7.
$y = \int_{\sqrt{x}}^{1} \cos(t^2) \, dt$
A. $\frac{-\cos x}{2\sqrt{x}}$
B. $\frac{\cos x}{2\sqrt{x}}$
C. $-\sin x$
D. $\cos x$
8.
$x$
A. $e$
B. $1$
C. $0$
D. $e-1$
9.
$F(x) = \int_{0}^{x} f(t) dt$
A. $2$
B. $4$
C. $3$
D. $1$
10.
$\frac{d}{dx} \int_{1}^{x^3} \ln t dt$
A. $9e^2$
B. $3e^2$
C. $e^3$
D. $1$

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Second fundamental theorem of integral calculus

Subtopic

Second fundamental theorem of integral calculus under Integrals for Grade 12 CBSE.

About Topic & Revision

Preview questions (no answers)

1.
$\int_{0}^{\pi/2} (\sin x + \cos x) \, dx$
A. $1$
B. $0$
C. $2$
D. $\pi$
2.
$\int_{0}^{1} (x^{2} + 1) \, dx$
A. $\frac{2}{3}$
B. $\frac{4}{3}$
C. $2$
D. $1$
3.
$\int_{1}^{2} e^{-x} \, dx$
A. $\frac{e-1}{e^{2}}$
B. $\frac{1-e}{e^{2}}$
C. $e-e^{2}$
D. $\frac{1}{e}$
4.
$\int_{0}^{1/\sqrt{2}} \frac{1}{\sqrt{1-x^{2}}} \, dx$
A. $\frac{\pi}{2}$
B. $\frac{\pi}{3}$
C. $\frac{\pi}{6}$
D. $\frac{\pi}{4}$
5.
$\int_{1}^{2} \frac{\ln x}{x^{2}} dx$
A. $\frac{1-\ln 2}{2}$
B. $\frac{1+\ln 2}{2}$
C. $1-\ln 2$
D. $\frac{\ln 2}{2}$
6.
$\int_{0}^{1} \frac{1}{\sqrt{1+x}} dx$
A. $2\sqrt{2} - 2$
B. $\sqrt{2} - 1$
C. $2\sqrt{2}$
D. $2$
7.
$\int_{0}^{\pi/2} e^{x} \cos x dx$
A. $\frac{e^{\pi/2} + 1}{2}$
B. $\frac{e^{\pi/2} - 1}{2}$
C. $e^{\pi/2} - 1$
D. $1$
8.
$\int_{0}^{\pi/2} \frac{dx}{1 + \cot x}$
A. $\frac{\pi}{2}$
B. $\frac{\pi}{4}$
C. $\pi$
D. $\frac{\pi}{3}$
9.
$\int_{0}^{1} \frac{dx}{e^x + e^{-x}}$
A. $\tan^{-1} e - \frac{\pi}{4}$
B. $\tan^{-1} e + \frac{\pi}{4}$
C. $\tan^{-1} e$
D. $\frac{\pi}{4} - \tan^{-1} e$
10.
$\int_{0}^{\pi/2} \frac{\sin x \cos x}{(\sin x + \cos x)^2} dx$
A. $\frac{\pi}{4} - \frac{1}{2}$
B. $\frac{\pi}{2} - 1$
C. $\frac{\pi}{4} + \frac{1}{2}$
D. $\frac{\pi}{8}$

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Evaluation of Definite Integrals by Substitution

Subtopic

Evaluation of Definite Integrals by Substitution under Integrals for Grade 12 CBSE.

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1.
$\int_0^{\pi/2} \sin^2 x \cos x dx$
A. $\frac{1}{2}$
B. $\frac{1}{4}$
C. $\frac{1}{3}$
D. $1$
2.
$\int_0^1 \frac{x^2}{(1+x^3)^2} dx$
A. $\frac{1}{3}$
B. $\frac{1}{6}$
C. $\frac{1}{2}$
D. $\frac{1}{9}$
3.
$\int_1^e \frac{\sin(\ln x)}{x} dx$
A. $1 - \cos 1$
B. $\cos 1$
C. $1$
D. $\sin 1$
4.
$\int_0^{\pi/4} (1+\tan x) \sec^2 x dx$
A. $\frac{1}{2}$
B. $1$
C. $\frac{3}{2}$
D. $2$
5.
$\int_0^{\ln 2} \frac{e^{2x}}{1 + e^x} dx$
A. $1 + \ln \frac{3}{2}$
B. $1 + \ln \frac{2}{3}$
C. $\ln \frac{3}{2}$
D. $\ln \frac{2}{3}$
6.
$\int_0^1 \frac{e^x - e^{-x}}{e^x + e^{-x}} dx$
A. $\ln(e^2+1)$
B. $\ln(\frac{e^2+1}{2e})$
C. $\ln(\frac{e+1}{2})$
D. $\ln(\frac{e^2-1}{2e})$
7.
$\int_0^{\pi/2} \frac{\cos x}{9+\sin^2 x} dx$
A. $\frac{1}{3}\tan^{-1}(1/3)$
B. $\tan^{-1}(1/3)$
C. $\frac{1}{9}\tan^{-1}(1/3)$
D. $\frac{\pi}{12}$
8.
$\int_0^{\pi/2} \log(\tan x) dx$
A. $1$
B. $\frac{\pi}{2}$
C. $0$
D. $-1$
9.
$\int_0^1 \frac{x^2}{1+x^6} dx$
A. $\frac{\pi}{12}$
B. $\frac{\pi}{6}$
C. $\frac{\pi}{4}$
D. $\frac{\pi}{3}$
10.
$\int_0^1 \frac{x^4+1}{x^6+1} dx$
A. $\frac{\pi}{6}$
B. $\frac{\pi}{2}$
C. $\frac{\pi}{3}$
D. $\frac{\pi}{4}$

Download the worksheet for Integrals - Evaluation of Definite Integrals by Substitution to practice offline. It includes additional chapter-level practice questions.

Some Properties of Definite Integrals

Subtopic

Some Properties of Definite Integrals under Integrals for Grade 12 CBSE.

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1.
$\int_{0}^{2a} f(x) \, dx$
A. $0$
B. $\int_{0}^{a} f(x) \, dx$
C. $2 \int_{0}^{a} f(x) \, dx$
D. $f(a)$
2.
$\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x} + \sqrt{x}} \, dx$
A. $1/2$
B. $3/2$
C. $1$
D. $2$
3.
$\int_{-\pi/2}^{\pi/2} \cos x \, dx$
A. $0$
B. $1$
C. $2$
D. $-2$
4.
$\int_{0}^{\pi/2} \frac{\sin^{n} x}{\sin^{n} x + \cos^{n} x} \, dx$
A. $\pi/2$
B. $\pi/4$
C. $n\pi$
D. $0$
5.
$\int_{-1}^1 \log(x + \sqrt{x^2+1}) dx$
A. $1$
B. $0$
C. $\log 2$
D. $\pi$
6.
$\int_0^1 \frac{dx}{x + \sqrt{1-x^2}}$
A. $\frac{\pi}{2}$
B. $\frac{\pi}{4}$
C. $\pi$
D. $1$
7.
$\int_{-2}^2 |x + 1| dx$
A. $4$
B. $5$
C. $2$
D. $3$
8.
$I = \int_{0}^{\pi} \frac{x \sin 2x \sin(\frac{\pi}{2} \cos x)}{2x - \pi} dx$
A. $\frac{8}{\pi}$
B. $\frac{4}{\pi}$
C. $\frac{2}{\pi}$
D. $2$
9.
$\int_{0}^{\pi} |\cos x| dx$
A. $2$
B. $1$
C. $0$
D. $4$
10.
$\int_{-1}^{1} e^{|x|} dx$
A. $2(e-1)$
B. $2e$
C. $e-1$
D. $2(e+1)$

Download the worksheet for Integrals - Some Properties of Definite Integrals to practice offline. It includes additional chapter-level practice questions.

Integrals

Integration as inverse process of differentiation

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Integration of a variety of functions by substitution, by partial fractions and by parts

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Evaluation of simple integrals and problems based on them

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Definite integrals as a limit of a sum

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Fundamental Theorem of Calculus

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Basic properties of definite integrals and evaluation of definite integrals

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Some properties of indefinite integral

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Methods of Integration

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Integration using trigonometric identities

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Integrals of Some Particular Functions

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Integral of the type ∫ e^x(f(x) + f'(x)) dx

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Integrals of some more types

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Definite Integral

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Area function

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First fundamental theorem of integral calculus

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Second fundamental theorem of integral calculus

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Evaluation of Definite Integrals by Substitution

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Some Properties of Definite Integrals

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