What is the integral of 81e^{-9x}sin(9x) ?

The integral of 81e^{-9x}sin(9x) is 81(-(e^{-9x}cos(9x))/(18)-(e^{-9x}sin(9x))/(18))+C

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integral of 81e^{-9x}sin(9x)

\int 81 e^{- 9 x} sin (9 x) d x

81 (- \frac{e ^{- 9 x} cos ( 9 x )}{18} - \frac{e ^{- 9 x} sin ( 9 x )}{18}) + C

Solution steps

\int 81 e^{- 9 x} sin (9 x) d x

Take the constant out:

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

= 81 \cdot \int e^{- 9 x} sin (9 x) d x

Apply Integration By Parts

: - \frac{1}{9} e^{- 9 x} cos (9 x) - \int e^{- 9 x} cos (9 x) d x

= 81 (- \frac{1}{9} e^{- 9 x} cos (9 x) - \int e^{- 9 x} cos (9 x) d x)

Apply Integration By Parts

: \frac{1}{9} e^{- 9 x} sin (9 x) - \int - e^{- 9 x} sin (9 x) d x

= 81 (- \frac{1}{9} e^{- 9 x} cos (9 x) - (\frac{1}{9} e^{- 9 x} sin (9 x) - \int - e^{- 9 x} sin (9 x) d x))

Take the constant out:

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

= 81 (- \frac{1}{9} e^{- 9 x} cos (9 x) - (\frac{1}{9} e^{- 9 x} sin (9 x) - (- \int e^{- 9 x} sin (9 x) d x)))

Therefore

81 \cdot \int e^{- 9 x} sin (9 x) d x = 81 (- \frac{1}{9} e^{- 9 x} cos (9 x) - (\frac{1}{9} e^{- 9 x} sin (9 x) - (- \int e^{- 9 x} sin (9 x) d x)))

Isolate

\int e^{- 9 x} sin (9 x) d x

= 81 (- \frac{e ^{- 9 x} cos ( 9 x )}{18} - \frac{e ^{- 9 x} sin ( 9 x )}{18})

Add a constant to the solution

= 81 (- \frac{e ^{- 9 x} cos ( 9 x )}{18} - \frac{e ^{- 9 x} sin ( 9 x )}{18}) + C

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What is the integral of 81e^{-9x}sin(9x) ?
The integral of 81e^{-9x}sin(9x) is 81(-(e^{-9x}cos(9x))/(18)-(e^{-9x}sin(9x))/(18))+C