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  4. ∫ (3 sin x-2 cos x/4 cos x+5 sin x) d x=

Q. $\int \frac{3 \sin x-2 \cos x}{4 \cos x+5 \sin x} d x=$

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Solution:

$\int \frac{a \cos x+b \sin x}{c \cos x+d \sin x}=\frac{a c+b d}{c^{2}+d^{2}} x+\frac{a d-b c}{c^{2}+d^{2}} \log (c \cos x+d \sin x) $ (Short cut)
Required integral $=\int \frac{-2 \cos x+3 \sin x}{4 \operatorname{oos} x+5 \sin x}$
$=\frac{-8+15}{16+25} x+\frac{-10-12}{16+25} \log (4 \cos x+5 \sin x) $

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