What is the integral of (sin(5y))/(e^{4y)} ?

The integral of (sin(5y))/(e^{4y)} is -4/41 e^{-4y}sin(5y)-5/41 e^{-4y}cos(5y)+C

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integral of (sin(5y))/(e^{4y)}

\int \frac{sin ( 5 y )}{e ^{4 y}} d y

- \frac{4}{41} e^{- 4 y} sin (5 y) - \frac{5}{41} e^{- 4 y} cos (5 y) + C

Solution steps

\int \frac{sin ( 5 y )}{e ^{4 y}} d y

Apply Integration By Parts

= - \frac{1}{4} e^{- 4 y} sin (5 y) - \int - \frac{5}{4} e^{- 4 y} cos (5 y) d y

\int - \frac{5}{4} e^{- 4 y} cos (5 y) d y = - \frac{5}{164} (5 e^{- 4 y} sin (5 y) - 4 e^{- 4 y} cos (5 y))

= - \frac{1}{4} e^{- 4 y} sin (5 y) - (- \frac{5}{164} (5 e^{- 4 y} sin (5 y) - 4 e^{- 4 y} cos (5 y)))

Simplify

- \frac{1}{4} e^{- 4 y} sin (5 y) - (- \frac{5}{164} (5 e^{- 4 y} sin (5 y) - 4 e^{- 4 y} cos (5 y))) : - \frac{4}{41} e^{- 4 y} sin (5 y) - \frac{5}{41} e^{- 4 y} cos (5 y)

= - \frac{4}{41} e^{- 4 y} sin (5 y) - \frac{5}{41} e^{- 4 y} cos (5 y)

Add a constant to the solution

= - \frac{4}{41} e^{- 4 y} sin (5 y) - \frac{5}{41} e^{- 4 y} cos (5 y) + C

What is the integral of (sin(5y))/(e^{4y)} ?
The integral of (sin(5y))/(e^{4y)} is -4/41 e^{-4y}sin(5y)-5/41 e^{-4y}cos(5y)+C