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

The integral of e^{2x}cos(e^{2x}) is 1/2 sin(e^{2x})+C

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

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

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

Solution steps

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

Apply u-substitution

: \int e^{u} cos (e^{u}) \frac{1}{2} d u

= \int e^{u} cos (e^{u}) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{u} cos (e^{u}) d u

Apply u-substitution

: \int cos (v) d v

= \frac{1}{2} \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= \frac{1}{2} sin (v)

Substitute back

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

Add a constant to the solution

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

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\frac{d}{d x} (3^{4 x})

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What is the integral of e^{2x}cos(e^{2x}) ?
The integral of e^{2x}cos(e^{2x}) is 1/2 sin(e^{2x})+C