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Integrals Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Integrals Set-6

Practice Important MCQs for Integrals — Mathematics, Class 12.

Integrals are central to Class 12 Mathematics because they connect area/accumulation with powerful tools like substitution, symmetry, partial fractions, and parameter-based techniques. Mastering standard integral results and proving them using differentiation under the integral sign or series expansions is crucial for board exams and is also a strong base for competitive exams like JEE/NEET, where unseen integrals often reduce to these core patterns.

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Q1. Evaluate the integral $I=\displaystyle\int_{-2}^{2} (x^{3}+x)\,e^{x^{2}}\,dx$ .

(A)

$e^{4}-1$

(B)

$2\bigl(e^{4}-1\bigr)$

(C)

$0$

(D)

$4\bigl(e^{4}+1\bigr)$

Q2. Let $I=\displaystyle\int_{0}^{\pi}\frac{x\sin x}{1+\cos^{2}x}\,dx$ . The value of $I$ is

(A)

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

(B)

$\dfrac{\pi}{2}$

(C)

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

(D)

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

Q3. Evaluate $\displaystyle\int_{0}^{1}\frac{\arctan\sqrt{x}}{\sqrt{x}\,\bigl(1+x\bigr)}\,dx$ .

(A)

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

(B)

$\dfrac{\pi^{2}}{12}$

(C)

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

(D)

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

Q4. Consider the statements: A: The improper integral $\displaystyle\int_{0}^{\infty}\frac{\sin x}{x}\,dx$ converges. R: The function $f(x)=\dfrac{\sin x}{x}$ is absolutely integrable on $(0,\infty)$ .

(A)

Both A and R are true and R is a correct explanation of A.

(B)

Both A and R are true but R is not a correct explanation of A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q5. Let $a,b>-1$ . Evaluate $I(a,b)=\displaystyle\int_{0}^{1}\frac{x^{a}-x^{b}}{\ln x}\,dx$ .

(A)

$\dfrac{a-b}{(a+1)(b+1)}$

(B)

$\displaystyle\ln\!\left(\frac{a+1}{b+1}\right)$

(C)

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

(D)

$0$

Q6. Evaluate the definite integral $I=\displaystyle\int_0^1 \frac{3x^2}{1+x^3}\,dx$ .

(A)

$\dfrac{1}{2}$

(B)

$\ln 3$

(C)

$\ln 2$

(D)

$\dfrac{2}{3}$

Q7. Evaluate $I=\displaystyle\int_0^1 \frac{x^2}{\sqrt{1-x^2}}\,dx$ .

(A)

$\dfrac{1}{2}$

(B)

$\dfrac{\pi}{4}$

(C)

$\dfrac{1}{3}$

(D)

$\dfrac{\pi}{2}$

Q8. Evaluate $I=\displaystyle\int_0^{\pi/4}\ln\bigl(1+\tan x\bigr)\,dx$ .

(A)

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

(B)

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

(C)

$\dfrac{\pi}{8}$

(D)

$\dfrac{\pi}{4}\ln\bigl(1+\sqrt{2}\bigr)$

Q9. Assertion (A): For $\alpha>-1$ , $I(\alpha)=\displaystyle\int_0^1 \frac{x^\alpha-1}{\ln x}\,dx=\ln(\alpha+1)$ .
Reason (R): Differentiating $I(\alpha)$ with respect to $\alpha$ under the integral sign gives $\displaystyle\int_0^1 x^\alpha\,dx=\frac{1}{\alpha+1}$ , and integrating this identity with respect to $\alpha$ (using the value at $\alpha=0$ ) yields $\ln(\alpha+1)$ .

(A)

Both A and R are true but R is not the correct explanation of A.

(B)

A is false but R is true.

(C)

Both A and R are false.

(D)

Both A and R are true and R is the correct explanation of A.

Q10. Evaluate the improper integral $I=\displaystyle\int_0^{\infty}\frac{e^{-2x}-e^{-3x}}{x}\,dx$ .

(A)

$\ln\!\dfrac{2}{3}$

(B)

$\ln\!\dfrac{3}{2}$

(C)

$\dfrac{1}{6}$

(D)

$\ln\!\dfrac{5}{6}$

Q11. Evaluate the definite integral $I=\displaystyle\int_{0}^{\pi/2}\sin^3 x\,dx$ .

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{2}{3}$

(C)

$1$

(D)

$\dfrac{1}{2}$

Q12. Evaluate the integral $I=\displaystyle\int_{0}^{1} x^2\ln(1+x)\,dx$ .

(A)

$\dfrac{1}{3}\ln2-\dfrac{1}{6}$

(B)

$\dfrac{2}{3}\ln2-\dfrac{5}{9}$

(C)

$\dfrac{12\ln2-5}{9}$

(D)

$\dfrac{12\ln2-5}{18}$

Q13. For integer $n\ge 1$ , evaluate $I_n=\displaystyle\int_{0}^{\pi} x\sin(nx)\,dx$ .

(A)

$\dfrac{\pi(-1)^{\,n+1}}{n}$

(B)

$-\dfrac{\pi}{n}$

(C)

$0$

(D)

$\dfrac{\pi(-1)^{\,n}}{n}$

Q14. Evaluate the integral $I=\displaystyle\int_{0}^{\pi/2}\ln\bigl(1+\tan x\bigr)\,dx$ .

(A)

$\pi\ln2$

(B)

$\dfrac{\pi}{2}\ln2$

(C)

$\dfrac{\pi}{4}\ln2+G$

(D)

$\dfrac{\pi}{4}\ln2$

Q15. For $a>-1$ , evaluate $I(a)=\displaystyle\int_{0}^{1}\frac{x^a-1}{\ln x}\,dx$ .

(A)

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

(B)

$\ln(a+1)$

(C)

$0$

(D)

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

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

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