What is the integral of 5e^{5x}sin(e^{5x}) ?

The integral of 5e^{5x}sin(e^{5x}) is -cos(e^{5x})+C

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integral of 5e^{5x}sin(e^{5x})

\int 5 e^{5 x} sin (e^{5 x}) d x

- cos (e^{5 x}) + C

Solution steps

\int 5 e^{5 x} sin (e^{5 x}) d x

Take the constant out:

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

= 5 \cdot \int e^{5 x} sin (e^{5 x}) d x

Apply u-substitution

= 5 \cdot \int \frac{sin ( u )}{5} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int sin (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{5} (- cos (u))

Substitute back

u = e^{5 x}

= 5 \cdot \frac{1}{5} (- cos (e^{5 x}))

Simplify

5 \cdot \frac{1}{5} (- cos (e^{5 x})) : - cos (e^{5 x})

= - cos (e^{5 x})

Add a constant to the solution

= - cos (e^{5 x}) + C

What is the integral of 5e^{5x}sin(e^{5x}) ?
The integral of 5e^{5x}sin(e^{5x}) is -cos(e^{5x})+C