$\int \frac{c o s x - s i n x}{7 - 9 s i n 2 x} d x$

\frac{1}{24} l n \frac{(4 + 3 s i n x + 3 c o s x)}{(4 - 3 s i n x - 3 c o s x)} + c

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\frac{1}{24} l n \frac{(4 - 3 c o s x - 3 s i n x)}{(4 - 3 c o s x + 3 s i n x)} + c

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\frac{1}{12} l n \frac{(4 + 3 s i n x - 3 c o s x)}{(4 - 3 s i n x + 3 c o s x)} + c

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\frac{1}{12} l n \frac{(4 + 3 c o s x + 3 s i n x)}{(4 - 3 s i n x - 3 c o s x)} + c

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Solution

The correct option is C $\frac{1}{24} l n \frac{(4 + 3 s i n x + 3 c o s x)}{(4 - 3 s i n x - 3 c o s x)} + c$
Let $I = \int \frac{cos x - sin x}{7 - 9 sin 2 x} d x = \int \frac{cos x - sin x}{16 - 9 {(sin x + cos x)}^{2}} d x$
Put $sin x + cos x = t \Rightarrow (cos x - sin x) d x = d t$
$I = \int \frac{1}{4^{2} - {(3 t)}^{2}} d t = \frac{1}{24} log (\frac{4 + 3 t}{4 - 3 t})$
$= \frac{1}{24} log (\frac{4 + 3 sin x + 3 cos x}{4 - 3 sin x - 3 cos x})$

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