Calculus

Integrals

Easy

common

$\displaystyle\int x\sqrt{x^2-1}\,dx$ is equal to : ( where C is an arbitrary constant )

$\dfrac{1}{\sqrt{x^2-1}}+C$

$\dfrac{2}{3}(x^2-1)^{3/2}+C$

$\dfrac{1}{3}(x^2-1)^{3/2}+C$

$\dfrac{-1}{\sqrt{x^2-1}}+C$

✅ Correct Option: 3

Substitute $t = x^2-1$ , giving $dt = 2x\,dx$ .

Then the integral is $\dfrac{1}{2}\int \sqrt{t}\,dt = \dfrac{1}{2}\cdot\dfrac{2}{3}t^{3/2} = \dfrac{1}{3}(x^2-1)^{3/2}+C$ .

2025: 2 June Shift 1

$\int e^{(x \log 5)}e^x dx$ , is:

Where $C$ is the constant of integration.

2026: 12 May Shift 2

Match the rational functions with the correct form of the partial fractions.

[ $p,q,r,a,b,c,A,B,C$ are arbitrary constants]

Match List-I with List-II

List-I (Rational Function)	List-II (Partial Fraction)
(A) $\dfrac{px^2+qx+r}{(x-a)(x^2+bx+c)}$	(I) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$
(B) $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}, a\neq b$	(II) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$
(C) $\dfrac{px^2+qx+r}{(x-a)(x-b)(x-c)}, a\neq b\neq c$	(III) $\dfrac{A}{x-a}+\dfrac{Bx+C}{x^2+bx+c}$
(D) $\dfrac{px+q}{(x-a)^2}$	(IV) $\dfrac{A}{x-a}+\dfrac{B}{x-b}+\dfrac{C}{x-c}$

Choose the correct answer from the options given below:

2022: 23 Aug Shift 1

$\int \frac{dx}{(e^x - 1)} =$

2025: 16 May Shift 1

The value of $\int_0^1 [\log x - \log(1-x)] dx$ is

2026: 30 May Shift 2

$\int e^{x\log_e a}.e^x dx$ is:

[where C is an arbitrary constant]

2026: 26 May Shift 2

The value of $\displaystyle\int_0^1 xe^{x^2}\,dx$ is equal to :

∫xx2−1 dx\displaystyle\int x\sqrt{x^2-1}\,dx∫xx2−1​dx is equal to : ( where C is an arbitrary constant )