e^x x (1 + x) dx equals : Option 1: e^x x + c Option 2: e^x x + c Option 3: e^x x + c Option 4: e^x x + c

Answer: e^x x + c. Solution: e^x[f(x) + f'(x)] dx = e^x f(x) + c. Here f(x) = x and f'(x) = x x, so the integral = e^x x + c.

Calculus

Integrals

Medium

core

$\int e^x \sec x (1 + \tan x) dx$ equals :

$e^x \sec x + c$

$e^x \tan x + c$

$e^x \sin x + c$

$e^x \cos x + c$

✅ Correct Option: 1

$\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + c$ . Here $f(x) = \sec x$ and $f'(x) = \sec x \tan x$ , so the integral $= e^x \sec x + c$ .

2025: 3 June Shift 1

Match List-I with List-II

List-I	List-II
Integral	Value
---	---
(A) $\int_{-1}^{1} (\vert x\vert + 1) dx$	(I) 0
(B) $\int_{-2}^{2} \vert x + 1\vert dx$	(II) 2
(C) $\int_{-1}^{1} 3\vert x^2\vert dx$	(III) 5
(D) $\int_{-1}^{1} x\vert x\vert dx$	(IV) 3

Choose the correct answer from the options given below:

2025: 15 May Shift 1

If $\int e^x\left(\frac{x-1}{(x+1)^3}\right)dx = \frac{Ae^x}{(x+1)^B} + C$ , where C is constant of integration then which of the following are correct?

(A) $A = -1$

(B) $A = 1$

(D) $B = 2$

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

2025: 22 May Shift 1

The Value of $\int_1^3 |2x - 1|dx$ equal to

2025: 22 May Shift 2

Match List-I with List-II (where $c$ is an arbitrary constant)

List-I	List-II
(A) $\int \tan x \, dx$	(I) $\log\vert \sec x + \tan x\vert + c$
(B) $\int \cot x \, dx$	(II) $\log\vert \sec x\vert + c$
(C) $\int \sec x \, dx$	(III) $\log\vert \sin x\vert + c$
(D) $\int \cosec x \, dx$	(IV) $\log\vert \cosec x - \cot x\vert + c$

Choose the correct answer from the options given below:

2025: 2 June Shift 1

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

Where $C$ is the constant of integration.

∫exsec⁡x(1+tan⁡x)dx\int e^x \sec x (1 + \tan x) dx∫exsecx(1+tanx)dx equals :

Solution

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$\int e^x \sec x (1 + \tan x) dx$ equals :