Question

tan^4 root x sec^2 root x / root x

Answer 1

GIVEN:- (tan⁴√x*sex²√x)/√x.dx

let √x = t

dt/dx = 1/2√x

dx/√x = 2dt

now, putting these values

∫(tan⁴t*sec²t)* 2dt

again put tant = p

dp/dt = sec²t

dp = sec²t*dt

therefore,

2∫tan⁴t*sec²t*dt = 2∫p⁴*dp

=> 2p⁵/5 + c

now substituting all the values

=> 2tan⁵√x/5 + c

where, c = arbitrary constant

Answer 2

Topic:

Integration

Solution:

An appropriate question is to evaluate the following integral.

$\displaystyle\int\dfrac{\tan^4\sqrt{x}\sec^2\sqrt{x}}{\sqrt{x}}\ dx$

We solve this problem using the method of substitution.

$\text{Put \tan\sqrt{x} = t so that \dfrac{\sec^2\sqrt{x}}{\sqrt x}\ dx = 2\ dt }$

The above integral changes to:

$\displaystyle\longrightarrow \int 2 t^4 \ dt$

Now, we will use the following identity:
$\boxed{\int x^n\ dx = \dfrac{x^{n+1}}{n+1} + C}$

Using this, we get:
$\displaystyle\longrightarrow 2 \cdot\dfrac{t^5}{5}\ +C$

Undoing our substitution for t, we get:
$\displaystyle\longrightarrow \dfrac{2\tan^5\sqrt{x}}{5}\ +C$

Hence the required answer is:

$\purple{\boxed{\displaystyle\int\dfrac{\tan^4\sqrt{x}\sec^2\sqrt{x}}{\sqrt{x}}\ dx = \dfrac{2\tan^5\sqrt{x}}{5}+ C}}$