int frac{d x}{(sin x+cos x)(2 cos x+sin x)} = | vedclass

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(D) Let $I = \int \frac{dx}{(\sin x + \cos x)(2 \cos x + \sin x)}$.Divide the numerator and denominator by $\cos^2 x$:$I = \int \frac{\sec^2 x}{(	an

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$\log \left|\frac{	an x+1}{	an x+2}ight|+c$

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(D) Let $I = \int \frac{dx}{(\sin x + \cos x)(2 \cos x + \sin x)}$.Divide the numerator and denominator by $\cos^2 x$:$I = \int \frac{\sec^2 x}{(	an x + 1)(2 + 	an x)} dx$.Let $	an x = t$,then $\sec^2 x dx = dt$.$I = \int \frac{dt}{(t+1)(t+2)}$.Using partial fractions: $\frac{1}{(t+1)(t+2)} = \frac{A}{t+1} + \frac{B}{t+2}$.$1 = A(t+2) + B(t+1)$.For $t = -1$,$A = 1$. For $t = -2$,$B = -1$.$I = \int \left( \frac{1}{t+1} - \frac{1}{t+2} ight) dt = \log|t+1| - \log|t+2| + C$.$I = \log \left| \frac{t+1}{t+2} ight| + C = \log \left| \frac{	an x + 1}{	an x + 2} ight| + C$.

$\int \frac{d x}{(\sin x+\cos x)(2 \cos x+\sin x)} = $

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