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

$y = c o s (8 x + 6) + c o s e c^{2} (7 x + 5) + 6 s e c x t a n x$

$\frac{s i n (8 x + 6)}{8} + \frac{c o t (7 x + 5)}{7} + 6 s e c x t a n x$

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$s i n (8 x) - c o t (7 x) + 6 s e c x + c$

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$\frac{s i n (8 x + 6)}{8} - \frac{c o t (7 x + 5)}{7} + 6 s e c x + c$

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$s i n (8 x + 6) - c o t (7 x + 5) 6 s e c x + c$

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Solution

The correct option is C
$\frac{s i n (8 x + 6)}{8} - \frac{c o t (7 x + 5)}{7} + 6 s e c x + c$

Let $u = 8 x + 6, \frac{d u}{d x} = 8 \Rightarrow d x = \frac{d u}{8}$

$t = 7 x + 5, \frac{d t}{d x} = 7 \Rightarrow d x = \frac{d t}{7}$

$\int y d x = \int c o s u \frac{d u}{8} + \int c o s e c^{2} t \frac{d t}{7} + 6 \int s e c x t a n x d x$

= $\frac{s i n u}{8} + \frac{(- c o t t)}{7} + 6 s e c x + c$

= $\frac{s i n (8 x + 6)}{8} - \frac{c o t (7 x + 5)}{7} + 6 s e c x + c$

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