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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int e^{c o s (u)} sin (u) \frac{1}{5} d u

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

Apply u-substitution

: \int - e^{v} d v

= \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 t)})

Simplify

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

Add a constant to the solution

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

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