First slide

Methods of integration

Question

$S = \int \frac{[\cot (\log x)]^{5} {cosec}^{2} (\log x)}{x} d x (x > 0) is equal to$

Moderate

A

$\frac{1}{12} {(\cot (\log x))}^{12} + C$
B

$\frac{- 1}{6} {(\cot (\log x))}^{6} + C$
C

$\frac{- 1}{6} {(\cot x)}^{6}$
D

$\frac{1}{16} {(\cot (\log x))}^{16} + C$

Solution

$S = \int \frac{{[\cot (\log x)]}^{5} \cos e c^{2} (\log x)}{x} d x$
$\log x = y$
$\begin{array}{l} \frac{1}{x} d x = d y \\ S = \int (\cot y)^{5} \cos e c^{2} (y) d y \end{array}$
$Put \cot y = z$
$\begin{array}{l} \cos e c^{2} y d y = - d z \\ S = \int z^{5} (- d z) \\ = \frac{- z^{6}}{6} + C \\ = - \frac{1}{6} (\cot y)^{6} + C \\ S = - \frac{1}{6} (\cot (\log x))^{6} + C \end{array}$

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