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

The integral of e^{2x}sin(e^{2x})cos(e^{2x}) is 1/4 sin^2(e^{2x})+C

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

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

\frac{1}{4} sin^{2} (e^{2 x}) + C

Solution steps

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

Apply u-substitution

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

= \int e^{u} sin (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} sin (e^{u}) cos (e^{u}) d u

Apply u-substitution

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

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

Apply u-substitution

: \int w d w

= \frac{1}{2} \cdot \int w d w

Apply the Power Rule

: \frac{w ^{2}}{2}

= \frac{1}{2} \cdot \frac{w ^{2}}{2}

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{sin ^{2} ( e ^{2 x} )}{2} : \frac{1}{4} sin^{2} (e^{2 x})

= \frac{1}{4} sin^{2} (e^{2 x})

Add a constant to the solution

= \frac{1}{4} sin^{2} (e^{2 x}) + C

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What is the integral of e^{2x}sin(e^{2x})cos(e^{2x}) ?
The integral of e^{2x}sin(e^{2x})cos(e^{2x}) is 1/4 sin^2(e^{2x})+C