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

The integral of cos(5x)e^{sin(5x)} is 1/5 e^{sin(5x)}+C

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

\int cos (5 x) e^{s i n (5 x)} d x

\frac{1}{5} e^{s i n (5 x)} + C

Solution steps

\int cos (5 x) e^{s i n (5 x)} d x

Apply u-substitution

= \int cos (u) e^{s i n (u)} \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int cos (u) e^{s i n (u)} d u

Apply u-substitution

= \frac{1}{5} \cdot \int e^{v} d v

Use the common integral:

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

= \frac{1}{5} e^{v}

Substitute back

= \frac{1}{5} e^{s i n (5 x)}

Add a constant to the solution

= \frac{1}{5} e^{s i n (5 x)} + C

What is the integral of cos(5x)e^{sin(5x)} ?
The integral of cos(5x)e^{sin(5x)} is 1/5 e^{sin(5x)}+C