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

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Question

(B) We have the integral $I = \int e^x \left( \frac{2 + \sin 2x}{1 + \cos 2x} \right) dx$. Using the trigonometric identities $\sin 2x = 2 \sin x \cos

Vedclass · Accepted Answer

$e^x 	an x + C$

Vedclass · Answer

(B) We have the integral $I = \int e^x \left( \frac{2 + \sin 2x}{1 + \cos 2x} ight) dx$. Using the trigonometric identities $\sin 2x = 2 \sin x \cos x$ and $1 + \cos 2x = 2 \cos^2 x$,we get: $I = \int e^x \left( \frac{2 + 2 \sin x \cos x}{2 \cos^2 x} ight) dx$ $I = \int e^x \left( \frac{2(1 + \sin x \cos x)}{2 \cos^2 x} ight) dx$ $I = \int e^x \left( \frac{1}{\cos^2 x} + \frac{\sin x \cos x}{\cos^2 x} ight) dx$ $I = \int e^x (\sec^2 x + 	an x) dx$. Recall the standard integral form $\int e^x (f(x) + f'(x)) dx = e^x f(x) + C$. Here,let $f(x) = 	an x$,then $f'(x) = \sec^2 x$. Therefore,$I = e^x 	an x + C$.

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

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