The value of ∫1sin3⁡xcos5⁡xdx, is - Infinity Learn by Sri Chaitanya

A
$\frac{- 2}{\sqrt{\tan x}} + \frac{2}{3} (\tan x)^{3 / 2} + C$
B
$\frac{2}{\sqrt{\tan x}} - \frac{2}{3} (\tan x)^{3 / 2} + C$
C
$\frac{- 2}{\sqrt{\tan x}} + \frac{2}{3} {(\tan x)}^{1 / 2} + C$
D
none of these

Solution:

We have,

$\begin{aligned} I = \int \frac{1}{\sqrt{\sin^{3} x \cos^{5} x}} d x \\ \Rightarrow I = \int \frac{1}{\sin^{3 / 2} x \cos^{\frac{5}{2}}} d x \end{aligned}$

$\Rightarrow I = \int \frac{\sec^{4} x}{\tan^{3 / 2} x} d x$ [Dividing $N^{r}$ and $D^{r}$ by $\cos^{2} x$ ]

$\begin{array}{l} \Rightarrow I = \int \frac{(1 + \tan^{2} x)}{\tan^{3 / 2} x} d (\tan x) \\ \Rightarrow I = \int \{(\tan x)^{- 3 / 2} + (\tan x)^{1 / 2}\} d (\tan x) \\ \Rightarrow I = - 2 (\tan x)^{- 1 / 2} + \frac{2}{3} (\tan x)^{3 / 2} + C \\ \Rightarrow I = \frac{- 2}{\sqrt{\tan x}} + \frac{2}{3} (\tan x)^{3 / 2} + C \end{array}$

The value of $\int \frac{1}{\sqrt{\sin^{3} x \cos^{5} x}} d x,$ is

Solution:

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