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

The integral of 8e^{-8x}sin(8x) is 8(-(e^{-8x}cos(8x))/(16)-(e^{-8x}sin(8x))/(16))+C

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

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

8 (- \frac{e ^{- 8 x} cos ( 8 x )}{16} - \frac{e ^{- 8 x} sin ( 8 x )}{16}) + C

Solution steps

\int 8 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

= 8 \cdot \int e^{- 8 x} sin (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

= 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

= 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

= 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

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

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

Add a constant to the solution

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

\frac{d y}{d x} + \frac{3}{x} y = x^{5}

y = \frac{1}{5 + 3 x}, (- 1, \frac{1}{2})

\int \frac{2 x + 3}{x ^{2} + 3 x - 10} d x

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

x \to \infty lim (\frac{1}{x} + 4)

What is the integral of 8e^{-8x}sin(8x) ?
The integral of 8e^{-8x}sin(8x) is 8(-(e^{-8x}cos(8x))/(16)-(e^{-8x}sin(8x))/(16))+C