What is the integral of 3e^{-5x}cos(e^{-5x}) ?

The integral of 3e^{-5x}cos(e^{-5x}) is -3/5 sin(e^{-5x})+C

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integral of 3e^{-5x}cos(e^{-5x})

\int 3 e^{- 5 x} cos (e^{- 5 x}) d x

- \frac{3}{5} sin (e^{- 5 x}) + C

Solution steps

\int 3 e^{- 5 x} cos (e^{- 5 x}) d x

Take the constant out:

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

= 3 \cdot \int e^{- 5 x} cos (e^{- 5 x}) d x

Apply u-substitution

: \int - \frac{1}{5} e^{u} cos (e^{u}) d u

= 3 \cdot \int - \frac{1}{5} e^{u} cos (e^{u}) d u

Take the constant out:

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

= 3 (- \frac{1}{5} \cdot \int e^{u} cos (e^{u}) d u)

Apply u-substitution

: \int cos (v) d v

= 3 (- \frac{1}{5} \cdot \int cos (v) d v)

Use the common integral:

\int cos (v) d v = sin (v)

= 3 (- \frac{1}{5} sin (v))

Substitute back

= 3 (- \frac{1}{5} sin (e^{- 5 x}))

Simplify

3 (- \frac{1}{5} sin (e^{- 5 x})) : - \frac{3}{5} sin (e^{- 5 x})

= - \frac{3}{5} sin (e^{- 5 x})

Add a constant to the solution

= - \frac{3}{5} sin (e^{- 5 x}) + C

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What is the integral of 3e^{-5x}cos(e^{-5x}) ?
The integral of 3e^{-5x}cos(e^{-5x}) is -3/5 sin(e^{-5x})+C