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} 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

Take the constant out:

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

= - \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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\int x I n^{3} x d x

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