int_0^{pi /2} frac{cos x}{(1 + sin x)(2 + sin | vedclass

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(A) Let $I = \int_0^{\pi /2} \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. When $x = 0$,$t = 0$. Whe

Vedclass · Accepted Answer

$\log \frac{4}{3}$

Vedclass · Answer

(A) Let $I = \int_0^{\pi /2} \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx$. Substitute $\sin x = t$,then $\cos x \,dx = dt$. When $x = 0$,$t = 0$. When $x = \pi/2$,$t = 1$. The integral becomes $I = \int_0^1 \frac{1}{(1 + t)(2 + t)} \,dt$. Using partial fractions,$\frac{1}{(1 + t)(2 + t)} = \frac{1}{1 + t} - \frac{1}{2 + t}$. Thus,$I = \int_0^1 \left( \frac{1}{1 + t} - \frac{1}{2 + t} \right) \,dt$. $I = [\log |1 + t| - \log |2 + t|]_0^1$. $I = [\log \frac{1 + t}{2 + t}]_0^1$. $I = \log \frac{2}{3} - \log \frac{1}{2} = \log \left( \frac{2/3}{1/2} \right) = \log \frac{4}{3}$.

$\int_0^{\pi /2} \frac{\cos x}{(1 + \sin x)(2 + \sin x)} \,dx = $

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