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

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(C) We are given the integral $I = \int e^{x} \left( \frac{\sin x + \cos x}{\cos^2 x} \right) dx$. Using the identity $1 - \sin^2 x = \cos^2 x$,we rew

Vedclass · Accepted Answer

$e^{x} \sec x + C$

Vedclass · Answer

(C) We are given the integral $I = \int e^{x} \left( \frac{\sin x + \cos x}{\cos^2 x} ight) dx$. Using the identity $1 - \sin^2 x = \cos^2 x$,we rewrite the expression: $I = \int e^{x} \left( \frac{\sin x}{\cos^2 x} + \frac{\cos x}{\cos^2 x} ight) dx$ $I = \int e^{x} (	an x \sec x + \sec x) dx$ Let $f(x) = \sec x$. Then $f'(x) = \sec x 	an x$. We know the standard integral formula $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$. Applying this to our integral: $I = \int e^{x} (\sec x + \sec x 	an x) dx = e^{x} \sec x + C$.

$\int e^{x} \left[ \frac{\sin x + \cos x}{\cos^2 x} \right] dx$ is equal to:

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