int frac{cos x}{(1 + sin x)(2 + sin x)} ,dx = | vedclass

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Question

(C) Let $I = \int \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. The integral becomes $I = \int \frac

Vedclass · Accepted Answer

$\log \left| \frac{1 + \sin x}{2 + \sin x} \right| + c$

Vedclass · Answer

(C) Let $I = \int \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. The integral becomes $I = \int \frac{dt}{(t + 1)(t + 2)}$. Using partial fractions,$\frac{1}{(t + 1)(t + 2)} = \frac{1}{t + 1} - \frac{1}{t + 2}$. Integrating both terms,$I = \int \frac{1}{t + 1} \,dt - \int \frac{1}{t + 2} \,dt = \log |t + 1| - \log |t + 2| + c$. Using the property $\log a - \log b = \log \frac{a}{b}$,we get $I = \log \left| \frac{t + 1}{t + 2} \right| + c$. Substituting $t = \sin x$ back,$I = \log \left| \frac{1 + \sin x}{2 + \sin x} \right| + c$.

$\int \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx = $

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