What is the integral of 8e^{8x}sin(e^{8x}) ?

The integral of 8e^{8x}sin(e^{8x}) is -cos(e^{8x})+C

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integral of 8e^{8x}sin(e^{8x})

\int 8 e^{8 x} sin (e^{8 x}) d x

- cos (e^{8 x}) + C

Solution steps

\int 8 e^{8 x} sin (e^{8 x}) d x

Take the constant out:

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

= 8 \cdot \int e^{8 x} sin (e^{8 x}) d x

Apply u-substitution

= 8 \cdot \int \frac{sin ( u )}{8} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int sin (u) d u

Use the common integral:

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

= 8 \cdot \frac{1}{8} (- cos (u))

Substitute back

u = e^{8 x}

= 8 \cdot \frac{1}{8} (- cos (e^{8 x}))

Simplify

8 \cdot \frac{1}{8} (- cos (e^{8 x})) : - cos (e^{8 x})

= - cos (e^{8 x})

Add a constant to the solution

= - cos (e^{8 x}) + C

What is the integral of 8e^{8x}sin(e^{8x}) ?
The integral of 8e^{8x}sin(e^{8x}) is -cos(e^{8x})+C