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

The integral of e^{2x}*cos(e^x) is e^xsin(e^x)+cos(e^x)+C

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integral of e^{2x}*cos(e^x)

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

e^{x} sin (e^{x}) + cos (e^{x}) + C

Solution steps

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

Apply u-substitution

: \int u cos (u) d u

= \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= u sin (u) - \int sin (u) d u

\int sin (u) d u = - cos (u)

= u sin (u) - (- cos (u))

Substitute back

u = e^{x}

= e^{x} sin (e^{x}) - (- cos (e^{x}))

Simplify

= e^{x} sin (e^{x}) + cos (e^{x})

Add a constant to the solution

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

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What is the integral of e^{2x}*cos(e^x) ?
The integral of e^{2x}*cos(e^x) is e^xsin(e^x)+cos(e^x)+C