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

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

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

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

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

Solution steps

\int 5 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

= 5 \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

= 5 (- \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

= 5 (- \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

= 5 (- \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

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

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

Add a constant to the solution

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

4 π (3 + x)^{2}

f (x) = \frac{x ^{2}}{x + 7}, at x = 1

\frac{d ^{3}}{d x ^{3}} ((2 x + 5)^{7})

y = (5 x + 4)^{2}

x^{3} - 5 x^{2}

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