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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int \frac{sin ( u )}{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 sin (u) d u

Use the common integral:

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

= \frac{1}{2} (- cos (u))

Substitute back

u = e^{2 x}

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

Simplify

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

Add a constant to the solution

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

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