What is the integral of 25e^{-5x}sin(5x) ?

The integral of 25e^{-5x}sin(5x) is 25(-(e^{-5x}cos(5x))/(10)-(e^{-5x}sin(5x))/(10))+C

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integral of 25e^{-5x}sin(5x)

\int 25 e^{- 5 x} sin (5 x) d x

25 (- \frac{e ^{- 5 x} cos ( 5 x )}{10} - \frac{e ^{- 5 x} sin ( 5 x )}{10}) + C

Solution steps

\int 25 e^{- 5 x} sin (5 x) d x

Take the constant out:

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

= 25 \cdot \int e^{- 5 x} sin (5 x) d x

Apply Integration By Parts

: - \frac{1}{5} e^{- 5 x} cos (5 x) - \int e^{- 5 x} cos (5 x) d x

= 25 (- \frac{1}{5} e^{- 5 x} cos (5 x) - \int e^{- 5 x} cos (5 x) d x)

Apply Integration By Parts

: \frac{1}{5} e^{- 5 x} sin (5 x) - \int - e^{- 5 x} sin (5 x) d x

= 25 (- \frac{1}{5} e^{- 5 x} cos (5 x) - (\frac{1}{5} e^{- 5 x} sin (5 x) - \int - e^{- 5 x} sin (5 x) d x))

Take the constant out:

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

= 25 (- \frac{1}{5} e^{- 5 x} cos (5 x) - (\frac{1}{5} e^{- 5 x} sin (5 x) - (- \int e^{- 5 x} sin (5 x) d x)))

Therefore

25 \cdot \int e^{- 5 x} sin (5 x) d x = 25 (- \frac{1}{5} e^{- 5 x} cos (5 x) - (\frac{1}{5} e^{- 5 x} sin (5 x) - (- \int e^{- 5 x} sin (5 x) d x)))

Isolate

\int e^{- 5 x} sin (5 x) d x

= 25 (- \frac{e ^{- 5 x} cos ( 5 x )}{10} - \frac{e ^{- 5 x} sin ( 5 x )}{10})

Add a constant to the solution

= 25 (- \frac{e ^{- 5 x} cos ( 5 x )}{10} - \frac{e ^{- 5 x} sin ( 5 x )}{10}) + C

\int - \frac{x}{1 - x ^{2}} d x

\frac{d}{d x} (x, (x))

x \to - 2 lim (x^{2} - a)

\frac{d y}{d x} = x^{10 c o s (x)}

What is the integral of 25e^{-5x}sin(5x) ?
The integral of 25e^{-5x}sin(5x) is 25(-(e^{-5x}cos(5x))/(10)-(e^{-5x}sin(5x))/(10))+C