$\int \frac{5 cos x - 3 sin x}{3 cos x + 5 sin x} d x$ is equal to

\frac{1}{3 cos x + 5 sin x} + C

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\frac{- 1}{(3 cos x + 5 sin x)^{2}} + C

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ln | 5 cos x - 3 sin x | + C

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ln | 3 cos x + 5 sin x | + C

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Solution

The correct option is D $ln | 3 cos x + 5 sin x | + C$
$I = \int \frac{5 cos x - 3 sin x}{3 cos x + 5 sin x} d x$
Put $3 cos x + 5 sin x = t$
$\Rightarrow (- 3 sin x + 5 cos x) d x = d t ∴ I = \int \frac{1}{t} d t = ln | t | + C = ln | 3 cos x + 5 sin x | + C$

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