First slide

Methods of integration

Question

$\int \frac{{(\cot \sqrt{\tan^{- 1} x})}^{4} \cos e c^{2} \sqrt{\tan^{- 1} x} d x}{(1 + x^{2}) \sqrt{\tan^{- 1} x}} =$

Moderate

A

$5 \cot^{5} \sqrt{\cot^{- 1} x} + C$
B

$- 5 \cot^{5} \sqrt{\cot^{- 1} x} + C$
C

$\frac{1}{5} \cot^{5} \sqrt{\tan^{- 1} x} + C$
D

$\frac{- 1}{5} \cot^{5} \sqrt{\tan^{- 1} x} + C$

Solution

$\frac{T = \int {[\cot \sqrt{\tan^{- 1} x}]}^{4} \cos e c^{2} \sqrt{\tan^{- 1} x} d x}{(1 + x^{2}) \sqrt{\tan^{- 1} x}}$
$Put \sqrt{\tan^{- 1} x} = y$
$T = \int {(\cot y)}^{4} \cos e c^{2} y d x$
$Put \cot y = z$
$\begin{array}{l} \cos e c^{2} y d y = - d z \\ T = \int z^{4} (- d z) \\ = \frac{- z^{5}}{5} + C \\ = \frac{- 1}{5} (\cot y)^{5} + C \\ = \frac{- 1}{5} \cot^{5} \sqrt{\tanh - 1 x} + C \end{array}$

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