int frac{operatorname{cosec} x}{3 cos x+4 sin | vedclass

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Question

(B) Let $I = \int \frac{\operatorname{cosec} x}{3 \cos x + 4 \sin x} dx$.Divide the numerator and denominator by $\sin x$:$I = \int \frac{\operatornam

Vedclass · Accepted Answer

$\frac{1}{3} \log \left|\frac{\sin x}{3 \cos x+4 \sin x}\right|+c$

Vedclass · Answer

(B) Let $I = \int \frac{\operatorname{cosec} x}{3 \cos x + 4 \sin x} dx$.Divide the numerator and denominator by $\sin x$:$I = \int \frac{\operatorname{cosec}^2 x}{3 \cot x + 4} dx$.Let $t = 3 \cot x + 4$.Then $dt = -3 \operatorname{cosec}^2 x dx$,which implies $\operatorname{cosec}^2 x dx = -\frac{1}{3} dt$.Substituting these into the integral:$I = \int \frac{-1/3}{t} dt = -\frac{1}{3} \ln |t| + C$.Substituting back $t = 3 \cot x + 4$:$I = -\frac{1}{3} \ln |3 \cot x + 4| + C = -\frac{1}{3} \ln \left| \frac{3 \cos x + 4 \sin x}{\sin x} \right| + C$.Using the property $-\ln |x| = \ln |1/x|$:$I = \frac{1}{3} \ln \left| \frac{\sin x}{3 \cos x + 4 \sin x} \right| + C$.

$\int \frac{\operatorname{cosec} x}{3 \cos x+4 \sin x} d x=$

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