What is the integral of (sin(8x)*e^{-8x})/8 ?

The integral of (sin(8x)*e^{-8x})/8 is 1/128 (-e^{-8x}cos(8x)-e^{-8x}sin(8x))+C

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integral of (sin(8x)*e^{-8x})/8

\int \frac{sin ( 8 x ) \cdot e ^{- 8 x}}{8} d x

\frac{1}{128} (- e^{- 8 x} cos (8 x) - e^{- 8 x} sin (8 x)) + C

Solution steps

\int \frac{sin ( 8 x ) e ^{- 8 x}}{8} d x

Take the constant out:

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

= \frac{1}{8} \cdot \int sin (8 x) e^{- 8 x} d x

Apply Integration By Parts

: - \frac{1}{8} e^{- 8 x} cos (8 x) - \int e^{- 8 x} cos (8 x) d x

= \frac{1}{8} (- \frac{1}{8} e^{- 8 x} cos (8 x) - \int e^{- 8 x} cos (8 x) d x)

Apply Integration By Parts

: \frac{1}{8} e^{- 8 x} sin (8 x) - \int - e^{- 8 x} sin (8 x) d x

= \frac{1}{8} (- \frac{1}{8} e^{- 8 x} cos (8 x) - (\frac{1}{8} e^{- 8 x} sin (8 x) - \int - e^{- 8 x} sin (8 x) d x))

Take the constant out:

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

= \frac{1}{8} (- \frac{1}{8} e^{- 8 x} cos (8 x) - (\frac{1}{8} e^{- 8 x} sin (8 x) - (- \int e^{- 8 x} sin (8 x) d x)))

Therefore

\frac{1}{8} \cdot \int e^{- 8 x} sin (8 x) d x = \frac{1}{8} (- \frac{1}{8} e^{- 8 x} cos (8 x) - (\frac{1}{8} e^{- 8 x} sin (8 x) - (- \int e^{- 8 x} sin (8 x) d x)))

Isolate

\int e^{- 8 x} sin (8 x) d x

= \frac{1}{8} (- \frac{e ^{- 8 x} cos ( 8 x )}{16} - \frac{e ^{- 8 x} sin ( 8 x )}{16})

Simplify

\frac{1}{8} (- \frac{e ^{- 8 x} cos ( 8 x )}{16} - \frac{e ^{- 8 x} sin ( 8 x )}{16}) : \frac{1}{128} (- e^{- 8 x} cos (8 x) - e^{- 8 x} sin (8 x))

= \frac{1}{128} (- e^{- 8 x} cos (8 x) - e^{- 8 x} sin (8 x))

Add a constant to the solution

= \frac{1}{128} (- e^{- 8 x} cos (8 x) - e^{- 8 x} sin (8 x)) + C

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What is the integral of (sin(8x)*e^{-8x})/8 ?
The integral of (sin(8x)*e^{-8x})/8 is 1/128 (-e^{-8x}cos(8x)-e^{-8x}sin(8x))+C