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Integrals Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Integrals Set-5

Practice Important MCQs for Integrals — Mathematics, Class 12.

Integrals are a core theme in Class 12 Mathematics and form the backbone of many problems in board exams as well as competitive tests like JEE/NEET. Mastery of key techniques—substitution, symmetry, transformations, and handling definite/improper integrals—helps you solve both standard and high-difficulty questions quickly and accurately.

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20

Questions

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Q1. Evaluate the definite integral

\int_0^1 x e^{x^2}\,dx

.

(A)

$\dfrac{e+1}{2}$

(B)

$\dfrac{e}{2}$

(C)

$\dfrac{e-1}{2}$

(D)

$\dfrac{1}{2}$

Q2. Evaluate

\int_{0}^{\ln 2}\frac{e^{2x}}{1+e^x}\,dx

.

(A)

$1-\ln\frac{3}{2}$

(B)

$\ln\frac{3}{2}$

(C)

$\ln\frac{4}{3}$

(D)

$1-\ln 2$

Q3. Evaluate the integral

\int_0^1 \frac{\arctan x}{1+x^2}\,dx

.

(A)

$\dfrac{\pi^2}{16}$

(B)

$\dfrac{\pi}{8}$

(C)

$\dfrac{1}{4}$

(D)

$\dfrac{\pi^2}{32}$

Q4. For the real parameter $a>-1$ , evaluate

\dfrac{1}{4}

.

(A)

$\ln(1+a^2)$

(B)

$\ln(1+a)$

(C)

$\dfrac{a}{1+a}$

(D)

$\dfrac{1}{1+a}$

Q5. Evaluate the integral

\int_0^1 \frac{\arctan\sqrt{x}}{\sqrt{x}\,(1+x)}\,dx

.

(A)

$\dfrac{\pi^2}{32}$

(B)

$\dfrac{\pi^2}{8}$

(C)

$\dfrac{\pi^2}{16}$

(D)

$\dfrac{\pi}{4}\ln 2$

Q6. Evaluate the integral $\displaystyle \int_0^1 \frac{x^2}{\sqrt{1-x^2}}\,dx$ .

(A)

$\dfrac{1}{2}$

(B)

$\dfrac{1}{3}$

(C)

$\dfrac{\pi}{4}$

(D)

$\dfrac{\pi}{6}$

Q7. Evaluate the integral $\displaystyle \int_0^1 \frac{\ln(1+x)}{1+x^2}\,dx$ .

(A)

$\dfrac{\pi}{8}\ln 2$

(B)

$\dfrac{\pi}{4}\ln 2$

(C)

$\dfrac{\pi}{16}\ln 2$

(D)

$\dfrac{\ln 2}{2}$

Q8. Compute the improper integral $\displaystyle \int_0^{\infty} \frac{x}{1+x^4}\,dx$ .

(A)

$\dfrac{\pi}{2}$

(B)

$\dfrac{\pi}{8}$

(C)

$1$

(D)

$\dfrac{\pi}{4}$

Q9. For real $p>-1$ , evaluate $I(p)=\displaystyle \int_0^1 \frac{x^p-1}{\ln x}\,dx$ .

(A)

$\dfrac{p}{p+1}$

(B)

$\ln(p+1)$

(C)

$\dfrac{1}{p+1}$

(D)

$\ln(1+p^2)$

Q10. Evaluate the improper integral $\displaystyle \int_0^{\infty} \frac{\ln x}{1+x^2}\,dx$ .

(A)

$0$

(B)

$\dfrac{\pi}{2}\ln 2$

(C)

$-\dfrac{\pi}{2}\ln 2$

(D)

$\dfrac{\pi}{4}\ln 2$

Q11. Evaluate the integral

\int_{0}^{1} x e^{x^{2}}\,dx

.

(A)

$e-1$

(B)

$\dfrac{e^{2}-1}{4}$

(C)

$\dfrac{1}{2}(e-1)$

(D)

$1$

Q12. Evaluate the integral

\int_{0}^{\pi/4} \ln\bigl(1+\tan x\bigr)\,dx

.

(A)

$\dfrac{\pi}{8}\ln 2$

(B)

$\dfrac{\pi}{4}\ln 2$

(C)

$\dfrac{\pi}{8}\ln 3$

(D)

$\dfrac{\pi}{16}\ln 2$

Q13. Evaluate the improper integral

\int_{0}^{\infty}\frac{\ln\bigl(1+x^{2}\bigr)}{1+x^{2}}\,dx

.

(A)

$\dfrac{\pi^{2}}{6}$

(B)

$\pi$

(C)

$2\ln 2$

(D)

$\pi\ln 2$

Q14. Evaluate the definite integral

\int_{0}^{1}\frac{\arctan x}{1+x^{2}}\,dx

.

(A)

$\dfrac{\pi}{8}\ln 2$

(B)

$\dfrac{\pi^{2}}{32}$

(C)

$\dfrac{\pi}{4}\ln 2$

(D)

$\dfrac{\pi^{2}}{24}$

Q15. For real $a>-1$ , evaluate the integral

I(a)=\int_{0}^{1}\frac{x^{a}-1}{\ln x}\,dx

.

(A)

$\ln(a+1)$

(B)

$\dfrac{a}{a+1}$

(C)

$\ln(1+a^2)$

(D)

$\dfrac{\ln(a+1)}{a}$

...and 5 more challenging questions available in the interactive simulator.

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