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

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

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

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

- \frac{1}{5} e^{c o s (5 x)} + C

Solution steps

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

Apply u-substitution

= \int sin (u) e^{c o s (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 sin (u) e^{c o s (u)} d u

Apply u-substitution

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

Take the constant out:

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

= \frac{1}{5} (- \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^{c o s (5 x)})

Simplify

= - \frac{1}{5} e^{c o s (5 x)}

Add a constant to the solution

= - \frac{1}{5} e^{c o s (5 x)} + C

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