_(-2)^2 x^2/(1+5^x) dx is equal to: Option 1: 8 Option 2: 3/8 Option 3: 8/3 Option 4: 2/3

Answer: 8/3. Solution: Let I=_(-2)^2x^2/(1+5^x)dx. Replacing x by -x gives I=_(-2)^2(x^2 5^x)/(1+5^x)dx. Adding, 2I=_(-2)^2x^2dx=16/3, so I=83.

Calculus

Integrals

Medium

core

$\displaystyle\int_{-2}^{2} \dfrac{x^2}{1+5^x}\, dx$ is equal to:

3/8

8/3

2/3

✅ Correct Option: 3

Let $I=\int_{-2}^{2}\dfrac{x^2}{1+5^x}dx$ . Replacing $x$ by $-x$ gives $I=\int_{-2}^{2}\dfrac{x^2 5^x}{1+5^x}dx$ .

Adding, $2I=\int_{-2}^{2}x^2dx=\dfrac{16}{3}$ , so $I=\dfrac83$ .

2025: 14 May Shift 2

$\int \frac{e^x(1 + x)dx}{\cos^2(e^x x)}$ is equal to

2025: 15 May Shift 2

Match List-I with List-II

[.] denotes the greatest integer function.

Choose the correct answer from the options given below:

2026: 22 May Shift 1

Match List-I with List-II

Where C is arbitrary constant.

List-I	List-II
(A) $\displaystyle\int \dfrac{dx}{x^2 - a^2} =$	(I) $\log\left\vert x + \sqrt{x^2+a^2}\right\vert + C$
(B) $\displaystyle\int \dfrac{dx}{a^2 - x^2} =$	(II) $\log\left\vert x + \sqrt{x^2-a^2}\right\vert + C$
(C) $\displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}} =$	(III) $\dfrac{1}{2a}\log\left\vert \dfrac{x-a}{x+a}\right\vert + C$
(D) $\displaystyle\int \dfrac{dx}{\sqrt{x^2 + a^2}} =$	(IV) $\dfrac{1}{2a}\log\left\vert \dfrac{a+x}{a-x}\right\vert + C$

Choose the correct answer from the options given below:

2025: 29 May Shift 2

Match List-I with List-II

List-I	List-II
Integral	Solution: C is an arbitrary constant
---	---
(A) $\int \frac{dx}{x^2 + 25}$	(I) $\frac{1}{10} \log \left\vert \frac{5 + x}{5 - x} \right\vert + C$
(B) $\int \frac{dx}{x^2 - 25}$	(II) $\log \vert x + \sqrt{x^2 - 25}\vert + C$
(C) $\int \frac{dx}{25 - x^2}$	(III) $\frac{1}{5} \tan^{-1} \left( \frac{x}{5} \right) + C$
(D) $\int \frac{dx}{\sqrt{x^2 - 25}}$	(IV) $\frac{1}{10} \log \left\vert \frac{x - 5}{x + 5} \right\vert + C$

Choose the correct answer from the options given below:

2026: 19 May Shift 1

The value of $\int_{0}^{3} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}}\,dx$ is

2026: 18 May Shift 1

If $\int \frac{1}{\tan x + \cot x}dx = k\cos 2x + C$ , where C is an arbitrary constant, then the value of $k$ is