What is the integral of cos(ax)e^{sin(ax)} ?

The integral of cos(ax)e^{sin(ax)} is 1/a e^{sin(ax)}+C

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integral of cos(ax)e^{sin(ax)}

\int cos (a x) e^{s i n (a x)} d x

\frac{1}{a} e^{s i n (a x)} + C

Solution steps

\int cos (a x) e^{s i n (a x)} d x

Apply u-substitution

: \int \frac{e ^{s i n (u)} cos ( u )}{a} d u

= \int \frac{e ^{s i n (u)} cos ( u )}{a} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int e^{s i n (u)} cos (u) d u

Apply u-substitution

: \int e^{v} d v

= \frac{1}{a} \cdot \int e^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= \frac{1}{a} e^{v}

Substitute back

= \frac{1}{a} e^{s i n (a x)}

Add a constant to the solution

= \frac{1}{a} e^{s i n (a x)} + C

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What is the integral of cos(ax)e^{sin(ax)} ?
The integral of cos(ax)e^{sin(ax)} is 1/a e^{sin(ax)}+C