What is the integral of e^{3x}sin(e^{3x}) ?

The integral of e^{3x}sin(e^{3x}) is -1/3 cos(e^{3x})+C

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integral of e^{3x}sin(e^{3x})

\int e^{3 x} sin (e^{3 x}) d x

- \frac{1}{3} cos (e^{3 x}) + C

Solution steps

\int e^{3 x} sin (e^{3 x}) d x

Apply u-substitution

= \int \frac{sin ( u )}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{3} (- cos (u))

Substitute back

u = e^{3 x}

= \frac{1}{3} (- cos (e^{3 x}))

Simplify

= - \frac{1}{3} cos (e^{3 x})

Add a constant to the solution

= - \frac{1}{3} cos (e^{3 x}) + C

What is the integral of e^{3x}sin(e^{3x}) ?
The integral of e^{3x}sin(e^{3x}) is -1/3 cos(e^{3x})+C