First slide

Methods of integration

Question

$\int \sin^{- 7 / 5} {xcos}^{- 3 / 5} dx$ is equal to

Moderate

A

$\frac{- 2}{5} (\tan x)^{2 / 5} + C$
B

$- \frac{5}{2} (\cot x)^{2 / 5} + C$
C

$\frac{- 5}{2} (\tan x)^{2 / 5} + C$
D

None of these

Solution

Here,
$p = - \frac{7}{5}, q = - \frac{3}{5}$
$\begin{array}{l} p + q = - 2 \\ I = \int \sin^{- 7 / 5} {xcos}^{- 3 / 5} xdx = \int \frac{\cos^{- 3 / 5} x}{\sin^{- 3 / 5} {xsin}^{2} x} dx \\ = \int (\cot x)^{- 3 / 5} {cosec}^{2} xdx \end{array}$
Put, $\cot x = t \Rightarrow {cosec}^{2} xdx = - dt$
So, $I = - \int t^{- 3 / 5} dt = - \frac{5}{2} (\cot x)^{2 / 5} + C$

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