(1- x)/(^2 x) dx is equal to: (where c is an arbitrary constant) Option 1: x - x + c Option 2: x - x + c Option 3: x - x + c Option 4: x - x + c

Answer: x - x + c. Solution: Split the integrand as ^2 x - x x. Integrating term by term gives x - x + c.

Calculus

Integrals

Easy

core

$\displaystyle\int \dfrac{1-\sin x}{\cos^2 x}\,dx$ is equal to:

(where $c$ is an arbitrary constant)

$\cot x - \cosec x + c$

$\cosec x - \cot x + c$

$\sec x - \tan x + c$

$\tan x - \sec x + c$

✅ Correct Option: 4

Split the integrand as $\sec^2 x - \sec x\tan x$ .

Integrating term by term gives $\tan x - \sec x + c$ .

2024: 16 May Shift 1

$\int \dfrac{\pi}{x^{n+1}-x} d x=$

2022: 4 Aug Shift 1

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$

2026: 15 May Shift 1

Which of the following statements are TRUE?

A. $\displaystyle\int_4^9 \dfrac{1}{\sqrt{x}}\, dx = 2$

B. $\displaystyle\int_{-4}^4 (x^{101} + x^{201} + x^{301})\, dx = 4\left(\dfrac{1}{102} + \dfrac{1}{202} + \dfrac{1}{302}\right)$

C. $\displaystyle\int_{-1}^1 x^{1012}\, dx = \dfrac{2}{1013}$

D. $\displaystyle\int (x^2 + 2x + 1)\, dx = \dfrac{(x+1)^3}{3} + c$ where $c$ is an arbitrary constant

Choose the correct answer from the options given below:

2026: 19 May Shift 2

$\int_1^4 \frac{\{x\}}{[x]} dx$ is equal to

[Where $\{x\}$ = fractional part function and $[x]$ = greatest integer function]

2026: 26 May Shift 2

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

2026: 15 May Shift 1

If C is an arbitrary constant, then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\displaystyle\int \sqrt{x^2+a^2}\, dx$	I.	$\dfrac{x}{2}\sqrt{a^2-x^2} + \dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a} + C$
B.	$\displaystyle\int \sqrt{a^2-x^2}\, dx$	II.	$\log\left\vert x + \sqrt{x^2+a^2}\right\vert + C$
C.	$\displaystyle\int \dfrac{1}{\sqrt{x^2+a^2}}\, dx$	III.	$\sin^{-1}\dfrac{x}{a} + C$
D.	$\displaystyle\int \dfrac{1}{\sqrt{a^2-x^2}}\, dx$	IV.	$\dfrac{x}{2}\sqrt{x^2+a^2} + \dfrac{a^2}{2}\log\left\vert x + \sqrt{x^2+a^2}\right\vert + C$

Choose the correct answer from the options given below: