Mathematics

Evaluate: $$\displaystyle\int \dfrac { \sec ^ { 8 } x } { cosec x } d x$$


SOLUTION
$$ I = \displaystyle\int \dfrac{\sec^{8}x}{cosec\,x}dx $$

$$ I =\displaystyle \int \dfrac{\sin\,x}{\cos^{8}x}dx $$

$$ u = \cos\,x \Rightarrow du = -\sin\,x\,dx $$

$$ \Rightarrow I =\displaystyle \int \dfrac{-du}{u^{8}} $$

$$ = \dfrac{u^{-7}}{7}+c $$

$$ = \dfrac{(\cos\,x)^{-7}}{7}+c $$

$$ = \dfrac{1}{7\cos^{7}x}+c $$
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Subjective Medium Published on 17th 09, 2020
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