What is the integral of 20cos(w)e^{-2w} ?

The integral of 20cos(w)e^{-2w} is 20((e^{-2w}sin(w))/5-(2e^{-2w}cos(w))/5)+C

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integral of 20cos(w)e^{-2w}

\int 20 cos (w) e^{- 2 w} d w

20 (\frac{e ^{- 2 w} sin ( w )}{5} - \frac{2 e ^{- 2 w} cos ( w )}{5}) + C

Solution steps

\int 20 cos (w) e^{- 2 w} d w

Take the constant out:

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

= 20 \cdot \int cos (w) e^{- 2 w} d w

Apply Integration By Parts

: e^{- 2 w} sin (w) - \int - 2 e^{- 2 w} sin (w) d w

= 20 (e^{- 2 w} sin (w) - \int - 2 e^{- 2 w} sin (w) d w)

Take the constant out:

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

= 20 (e^{- 2 w} sin (w) - (- 2 \cdot \int e^{- 2 w} sin (w) d w))

Apply Integration By Parts

: - e^{- 2 w} cos (w) - \int 2 e^{- 2 w} cos (w) d w

= 20 (e^{- 2 w} sin (w) - (- 2 (- e^{- 2 w} cos (w) - \int 2 e^{- 2 w} cos (w) d w)))

Take the constant out:

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

= 20 (e^{- 2 w} sin (w) - (- 2 (- e^{- 2 w} cos (w) - 2 \cdot \int e^{- 2 w} cos (w) d w)))

Therefore

20 \cdot \int e^{- 2 w} cos (w) d w = 20 (e^{- 2 w} sin (w) - (- 2 (- e^{- 2 w} cos (w) - 2 \cdot \int e^{- 2 w} cos (w) d w)))

Isolate

\int e^{- 2 w} cos (w) d w

= 20 (\frac{e ^{- 2 w} sin ( w )}{5} - \frac{2 e ^{- 2 w} cos ( w )}{5})

Add a constant to the solution

= 20 (\frac{e ^{- 2 w} sin ( w )}{5} - \frac{2 e ^{- 2 w} cos ( w )}{5}) + C

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What is the integral of 20cos(w)e^{-2w} ?
The integral of 20cos(w)e^{-2w} is 20((e^{-2w}sin(w))/5-(2e^{-2w}cos(w))/5)+C