Find the integral of the given function w.r.t x

$y = s i n 6 x + 10 s e c^{2} x - c o s e c x c o t x$

$6 c o s 6 x + 20 s e c^{2} x t a n x + c o s e c x c o t x + c o s e c^{3} x + c$

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$c o s 6 x + 10 t a n x + c o s e c x + c$

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$- \frac{c o s (6 x)}{6} + 10 t a n x - (c o s e c x) + c$

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$- c o s e c 6 x + 10 t a n x + c o s e c x + c$

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Solution

The correct option is C
$- \frac{c o s (6 x)}{6} + 10 t a n x - (c o s e c x) + c$

$y = s i n 6 x + 10 s e c^{2} x - c o s e c x c o t x$

$\Rightarrow \int (s i n 6 x + 10 s e c^{2} x - c o s e c x c o t x) d x$

$\Rightarrow \int (s i n 6 x) d x + \int (10 s e c^{2} x) d x - \int (c o s e c x c o t x) d x$

$6 x = t \Rightarrow 6 d x = d t \Rightarrow d x = \frac{d t}{6}$

$\Rightarrow \int \frac{(s i n t) d t}{6} + 10 t a n x + c_{1} - (- c o s e c x) + c_{2}$

$\Rightarrow \frac{- c o s (t)}{6} + c_{o} + 10 t a n x + c_{1} + c o s e c x + c_{2}$

$\Rightarrow \frac{- c o s 6 x}{6} + 10 t a n x + c o s e c x + c$

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