The value of int e^{x} left[ frac{1+sin x}{1+ | vedclass

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(D) Let $I = \int e^{x} \left[ \frac{1+\sin x}{1+\cos x} \right] dx$. Using the identities $1+\cos x = 2\cos^2 \frac{x}{2}$ and $\sin x = 2\sin \frac{

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$e^{x} 	an \frac{x}{2} + C$

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(D) Let $I = \int e^{x} \left[ \frac{1+\sin x}{1+\cos x} ight] dx$. Using the identities $1+\cos x = 2\cos^2 \frac{x}{2}$ and $\sin x = 2\sin \frac{x}{2} \cos \frac{x}{2}$,we get: $I = \int e^{x} \left[ \frac{1 + 2\sin \frac{x}{2} \cos \frac{x}{2}}{2\cos^2 \frac{x}{2}} ight] dx$ $I = \int e^{x} \left[ \frac{1}{2\cos^2 \frac{x}{2}} + \frac{2\sin \frac{x}{2} \cos \frac{x}{2}}{2\cos^2 \frac{x}{2}} ight] dx$ $I = \int e^{x} \left[ \frac{1}{2} \sec^2 \frac{x}{2} + 	an \frac{x}{2} ight] dx$. We know that $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$. Here,let $f(x) = 	an \frac{x}{2}$. Then $f'(x) = \sec^2 \frac{x}{2} \cdot \frac{1}{2} = \frac{1}{2} \sec^2 \frac{x}{2}$. Thus,the integral becomes $e^{x} 	an \frac{x}{2} + C$.

The value of $\int e^{x} \left[ \frac{1+\sin x}{1+\cos x} \right] dx$ is

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