∫cos4xdxsin3x(sin5x+cos5x)35=−12(1+cotAx)B+C then AB=

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\int \frac{c o s^{4} x d x}{s i n^{3} x (s i n^{5} x + c o s^{5} x)^{\frac{3}{5}}} = - \frac{1}{2} (1 + c o t^{A} x)^{B} + C

then AB=

Options:

A . 1

B . 2

C . 12

D . None of these

Answer: Option B
:
B

I = \int \frac{c o s x^{4} d x}{s i n^{3} x (s i n^{5} x + c o s^{5} x)^{\frac{3}{5}}} = \int \frac{c o s e c^{2} x c o t^{4} x d x}{(1 + c o t^{5} x)^{3 / 5}} 1 + c o t^{5} x = t . 5 c o t^{4} x (- c o s e c^{2} x) d x = d t = \frac{- 1}{5} \int \frac{d t}{t^{3 / 5}} = \frac{- t^{2 / 5}}{2} = \frac{- 1}{2} (1 + c o t^{5} x)^{2 / 5} + C .

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