$\int \frac{c o s^{3} x + c o s^{5} x}{s i n^{2} x + s i n^{4} x} d x$ equals

s i n x - 6 t a n^{- 1} (s i n x) + c

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s i n x - 2 s i n^{- 1} x + c

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s i n x - 2 (s i n x)^{- 1} - 6 t a n^{- 1} (s i n x) + c

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s i n x - 2 (s i n x)^{- 1} + 5 t a n^{- 1} (s i n x) + c

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Solution

The correct option is C $s i n x - 2 (s i n x)^{- 1} - 6 t a n^{- 1} (s i n x) + c$

$s i n x = t; I = \int \frac{(1 - t^{2}) (2 - t^{2})}{t^{2} (1 + t^{2})} d t = \int (1 + \frac{2}{t^{2}} - \frac{6}{1 + t^{2}}) d t$
$= s i n x - 2 (s i n x)^{- 1} - 6 t a n^{- 1} (s i n x) + c$

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