What is the integral of 5e^xcos^2(x) ?

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

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integral of 5e^xcos^2(x)

\int 5 e^{x} cos^{2} (x) d x

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

Solution steps

\int 5 e^{x} cos^{2} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int e^{x} cos^{2} (x) d x

Apply Integration By Parts

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

= 5 (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)

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

Simplify

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

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

Add a constant to the solution

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

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