int e^x left[ frac{2 + sin 2x}{1 + cos 2x} ri | vedclass

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(A) Given integral: $I = \int e^x \left[ \frac{2 + \sin 2x}{1 + \cos 2x} \right] dx$ Using trigonometric identities $\sin 2x = 2 \sin x \cos x$ and $1

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$e^x 	an x + C$

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(A) Given integral: $I = \int e^x \left[ \frac{2 + \sin 2x}{1 + \cos 2x} ight] dx$ Using trigonometric identities $\sin 2x = 2 \sin x \cos x$ and $1 + \cos 2x = 2 \cos^2 x$: $I = \int e^x \left[ \frac{2 + 2 \sin x \cos x}{2 \cos^2 x} ight] dx$ $I = \int e^x \left[ \frac{2(1 + \sin x \cos x)}{2 \cos^2 x} ight] dx$ $I = \int e^x \left[ \frac{1}{\cos^2 x} + \frac{\sin x \cos x}{\cos^2 x} ight] dx$ $I = \int e^x (\sec^2 x + 	an x) dx$ Since $\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$,where $f(x) = 	an x$ and $f'(x) = \sec^2 x$: $I = e^x 	an x + C$

$\int e^x \left[ \frac{2 + \sin 2x}{1 + \cos 2x} \right] dx =$

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