What is the integral of cos^2(x)e^x ?

The integral of cos^2(x)e^x is e^xcos^2(x)-2/5 e^xcos(2x)+1/5 e^xsin(2x)+C

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integral of cos^2(x)e^x

\int cos^{2} (x) e^{x} d x

e^{x} cos^{2} (x) - \frac{2}{5} e^{x} cos (2 x) + \frac{1}{5} e^{x} sin (2 x) + C

Solution steps

\int cos^{2} (x) e^{x} d x

Apply Integration By Parts

: e^{x} cos^{2} (x) - \int - e^{x} sin (2 x) d x

= e^{x} cos^{2} (x) - \int - e^{x} sin (2 x) d x

\int - e^{x} sin (2 x) d x = \frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x)

= e^{x} cos^{2} (x) - (\frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x))

Simplify

= e^{x} cos^{2} (x) - \frac{2}{5} e^{x} cos (2 x) + \frac{1}{5} e^{x} sin (2 x)

Add a constant to the solution

= e^{x} cos^{2} (x) - \frac{2}{5} e^{x} cos (2 x) + \frac{1}{5} e^{x} sin (2 x) + C

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What is the integral of cos^2(x)e^x ?
The integral of cos^2(x)e^x is e^xcos^2(x)-2/5 e^xcos(2x)+1/5 e^xsin(2x)+C