What is the integral of e^{4x}sin(e^{4x}) ?

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

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integral of e^{4x}sin(e^{4x})

\int e^{4 x} sin (e^{4 x}) d x

- \frac{1}{4} cos (e^{4 x}) + C

Solution steps

\int e^{4 x} sin (e^{4 x}) d x

Apply u-substitution

: \int \frac{sin ( u )}{4} d u

= \int \frac{sin ( u )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{4} (- cos (u))

Substitute back

u = e^{4 x}

= \frac{1}{4} (- cos (e^{4 x}))

Simplify

= - \frac{1}{4} cos (e^{4 x})

Add a constant to the solution

= - \frac{1}{4} cos (e^{4 x}) + C

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