(^5√x.^2√x)/√xdx is equal to: Option 1: 1/3^5√x+C, where C is an arbitrary constant Option 2: 1/3^6√x+C, where C is an arbitrary constant Option 3: 1/5^5√x+C, where C is an arbitrary constant Option 4: ^6√x+C, where C is an arbitrary constant

Answer: 1/3^6√x+C, where C is an arbitrary constant. Solution: Put t= x, so dt=(^2 x)/(2 x)dx. The integral becomes 2 t^5dt=t^6/3=13^6 x+C.

Calculus

Integrals

Medium

core

$\int\frac{\tan^5\sqrt{x}.\sec^2\sqrt{x}}{\sqrt{x}}dx$ is equal to:

$\frac{1}{3}\tan^5\sqrt{x}+C$ , where C is an arbitrary constant

$\frac{1}{3}\tan^6\sqrt{x}+C$ , where C is an arbitrary constant

$\frac{1}{5}\tan^5\sqrt{x}+C$ , where C is an arbitrary constant

$\tan^6\sqrt{x}+C$ , where C is an arbitrary constant

✅ Correct Option: 2

Put $t=\tan\sqrt x$ , so $dt=\frac{\sec^2\sqrt x}{2\sqrt x}dx$ . The integral becomes $2\int t^5dt=\frac{t^6}{3}=\frac13\tan^6\sqrt x+C$ .

2026: 19 May Shift 2

$\int 5^x dx$ is

(where C is an arbitrary constant)

2025: 14 May Shift 2

The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is:

2022: 6 Aug Shift 2

If $f'(x) = 4x^5 - 6x$ and $f(0) = 3$ , then $f(3)$ is equal to

2026: 18 May Shift 2

$\int 3x^2 e^{x^3}\,dx$ is equal to : (where $C$ is an arbitrary constant)

2026: 31 May Shift 2

Match the LIST-I with LIST-II (where $[x]$ denotes greatest integer function)

	LIST-I Integral		LIST-II Values
A.	$\displaystyle\int_{0}^{2} [x]\, dx$	I.	$0$
B.	$\displaystyle\int_{0}^{1.5} [x^2]\, dx$	II.	$1$
C.	$\displaystyle\int_{-\pi/4}^{\pi/4} \sin x \, dx$	III.	$\sqrt{2}$
D.	$\displaystyle\int_{-\pi/4}^{\pi/4} \cos x \, dx$	IV.	$2-\sqrt{2}$

Choose the correct answer from the options given below:

2025: 19 May Shift 1

If $2f(x) + f\left(\frac{1}{x}\right) = x^2 + 1$ , then $\int f(x) dx$ is: (Here C is an arbitrary constant)