What is the integral of e^{4x}cos(e^x) ?

The integral of e^{4x}cos(e^x) is e^{3x}sin(e^x)+3e^{2x}cos(e^x)-6e^xsin(e^x)-6cos(e^x)+C

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integral of e^{4x}cos(e^x)

\int e^{4 x} cos (e^{x}) d x

e^{3 x} sin (e^{x}) + 3 e^{2 x} cos (e^{x}) - 6 e^{x} sin (e^{x}) - 6 cos (e^{x}) + C

Solution steps

\int e^{4 x} cos (e^{x}) d x

Apply Integration By Parts

: \frac{1}{4} e^{4 x} cos (e^{x}) - \int - \frac{1}{4} e^{5 x} sin (e^{x}) d x

= \frac{1}{4} e^{4 x} cos (e^{x}) - \int - \frac{1}{4} e^{5 x} sin (e^{x}) d x

\int - \frac{1}{4} e^{5 x} sin (e^{x}) d x = \frac{1}{4} e^{4 x} cos (e^{x}) - e^{3 x} sin (e^{x}) - 3 e^{2 x} cos (e^{x}) + 6 e^{x} sin (e^{x}) + 6 cos (e^{x})

= \frac{1}{4} e^{4 x} cos (e^{x}) - (\frac{1}{4} e^{4 x} cos (e^{x}) - e^{3 x} sin (e^{x}) - 3 e^{2 x} cos (e^{x}) + 6 e^{x} sin (e^{x}) + 6 cos (e^{x}))

Simplify

\frac{1}{4} e^{4 x} cos (e^{x}) - (\frac{1}{4} e^{4 x} cos (e^{x}) - e^{3 x} sin (e^{x}) - 3 e^{2 x} cos (e^{x}) + 6 e^{x} sin (e^{x}) + 6 cos (e^{x})) : e^{3 x} sin (e^{x}) + 3 e^{2 x} cos (e^{x}) - 6 e^{x} sin (e^{x}) - 6 cos (e^{x})

= e^{3 x} sin (e^{x}) + 3 e^{2 x} cos (e^{x}) - 6 e^{x} sin (e^{x}) - 6 cos (e^{x})

Add a constant to the solution

= e^{3 x} sin (e^{x}) + 3 e^{2 x} cos (e^{x}) - 6 e^{x} sin (e^{x}) - 6 cos (e^{x}) + C

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What is the integral of e^{4x}cos(e^x) ?
The integral of e^{4x}cos(e^x) is e^{3x}sin(e^x)+3e^{2x}cos(e^x)-6e^xsin(e^x)-6cos(e^x)+C