What is the integral of 7e^{7x}sin(e^{7x}) ?

The integral of 7e^{7x}sin(e^{7x}) is -cos(e^{7x})+C

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integral of 7e^{7x}sin(e^{7x})

\int 7 e^{7 x} sin (e^{7 x}) d x

- cos (e^{7 x}) + C

Solution steps

\int 7 e^{7 x} sin (e^{7 x}) d x

Take the constant out:

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

= 7 \cdot \int e^{7 x} sin (e^{7 x}) d x

Apply u-substitution

: \int \frac{sin ( u )}{7} d u

= 7 \cdot \int \frac{sin ( u )}{7} d u

Take the constant out:

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

= 7 \cdot \frac{1}{7} \cdot \int sin (u) d u

Use the common integral:

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

= 7 \cdot \frac{1}{7} (- cos (u))

Substitute back

u = e^{7 x}

= 7 \cdot \frac{1}{7} (- cos (e^{7 x}))

Simplify

7 \cdot \frac{1}{7} (- cos (e^{7 x})) : - cos (e^{7 x})

= - cos (e^{7 x})

Add a constant to the solution

= - cos (e^{7 x}) + C

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What is the integral of 7e^{7x}sin(e^{7x}) ?
The integral of 7e^{7x}sin(e^{7x}) is -cos(e^{7x})+C