Match List-I with List-II, List-I, List-II, (A) The value of I=_0^1 1/√xdx is, (I) 1, (B) The value of I=_0^1 (5x^2+1)dx is, (II) 1/2, (C) The value of I=_0^∞ e^(-x)dx is, (III) 8/3, (D) The value of I=_1^e ( x)/xdx is, (IV) 2, Choose the correct answer from the options given below: Option 1: (A) - (I), (B) - (III), (C) - (IV), (D) - (II) Option 2: (A) - (IV), (B) - (III), (C) - (I), (D) - (II) Option 3: (A) - (IV), (B) - (I), (C) - (II), (D) - (III) Option 4: (A) - (II), (B) - (IV), (C) - (III), (D) - (I)

Answer: (A) - (IV), (B) - (III), (C) - (I), (D) - (II). Solution: (A) [2 x]_0^1=2, (IV). (B) [5x^3/3+x]_0^1=83, (III). (C) [-e^(-x)]_0^∞=1, (I). (D) Put t= x: _0^1t dt=12, (II).

Calculus

Integrals

Medium

applied

Match List-I with List-II

List-I List-II
(A) The value of $I=\int_0^1 \frac{1}{\sqrt{x}}dx$ is (I) 1
(B) The value of $I=\int_0^1 (5x^2+1)dx$ is (II) $\frac{1}{2}$
(C) The value of $I=\int_0^\infty e^{-x}dx$ is (III) $\frac{8}{3}$
(D) The value of $I=\int_1^e \frac{\log x}{x}dx$ is (IV) 2

Choose the correct answer from the options given below:

List-I	List-II
(A) The value of $I=\int_0^1 \frac{1}{\sqrt{x}}dx$ is	(I) 1
(B) The value of $I=\int_0^1 (5x^2+1)dx$ is	(II) $\frac{1}{2}$
(C) The value of $I=\int_0^\infty e^{-x}dx$ is	(III) $\frac{8}{3}$
(D) The value of $I=\int_1^e \frac{\log x}{x}dx$ is	(IV) 2

(A) - (I), (B) - (III), (C) - (IV), (D) - (II)

(A) - (IV), (B) - (III), (C) - (I), (D) - (II)

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

✅ Correct Option: 2

(A) $\left[2\sqrt x\right]_0^1=2$ , (IV).

(B) $\left[\frac{5x^3}{3}+x\right]_0^1=\frac83$ , (III).

(D) Put $t=\log x$ : $\int_0^1t\,dt=\frac12$ , (II).

2022: 16 July Shift 2

Match List I with List II

List I	List II
A. $\int \frac{dx}{x^2 - a^2}$ is equal to	I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$
B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to	II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$
C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to	III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$
D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to	IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$

Choose the correct answer from the options given below:

2025: 13 May Shift 1

The value of which of the following integrals is zero?

(A) $\int_0^1 x dx$

(B) $\int_{-1}^1 x dx$

(D) $\int_0^1 \log\left(\frac{x}{1-x}\right) dx$

Choose the correct answer from the options given below:

2025: 15 May Shift 2

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|) dx$ is

2026: 19 May Shift 1

The value of integral $I = \int_{3}^{5} \frac{x^2}{(x-1)(x-2)}\,dx$ is

2026: 22 May Shift 1

Value of $\int_{-2}^{2}|2x+1|dx$ is:

2025: 22 May Shift 1

$\int \frac{x}{(x-1)(x-2)} dx$ is equal to ( where $C$ is a constant of integration)

Solution

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