What is the integral of 7e^{-7x}sin(7x) ?

The integral of 7e^{-7x}sin(7x) is 7(-(e^{-7x}cos(7x))/(14)-(e^{-7x}sin(7x))/(14))+C

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integral of 7e^{-7x}sin(7x)

\int 7 e^{- 7 x} sin (7 x) d x

7 (- \frac{e ^{- 7 x} cos ( 7 x )}{14} - \frac{e ^{- 7 x} sin ( 7 x )}{14}) + C

Solution steps

\int 7 e^{- 7 x} sin (7 x) d x

Take the constant out:

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

= 7 \cdot \int e^{- 7 x} sin (7 x) d x

Apply Integration By Parts

: - \frac{1}{7} e^{- 7 x} cos (7 x) - \int e^{- 7 x} cos (7 x) d x

= 7 (- \frac{1}{7} e^{- 7 x} cos (7 x) - \int e^{- 7 x} cos (7 x) d x)

Apply Integration By Parts

: \frac{1}{7} e^{- 7 x} sin (7 x) - \int - e^{- 7 x} sin (7 x) d x

= 7 (- \frac{1}{7} e^{- 7 x} cos (7 x) - (\frac{1}{7} e^{- 7 x} sin (7 x) - \int - e^{- 7 x} sin (7 x) d x))

Take the constant out:

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

= 7 (- \frac{1}{7} e^{- 7 x} cos (7 x) - (\frac{1}{7} e^{- 7 x} sin (7 x) - (- \int e^{- 7 x} sin (7 x) d x)))

Therefore

7 \cdot \int e^{- 7 x} sin (7 x) d x = 7 (- \frac{1}{7} e^{- 7 x} cos (7 x) - (\frac{1}{7} e^{- 7 x} sin (7 x) - (- \int e^{- 7 x} sin (7 x) d x)))

Isolate

\int e^{- 7 x} sin (7 x) d x

= 7 (- \frac{e ^{- 7 x} cos ( 7 x )}{14} - \frac{e ^{- 7 x} sin ( 7 x )}{14})

Add a constant to the solution

= 7 (- \frac{e ^{- 7 x} cos ( 7 x )}{14} - \frac{e ^{- 7 x} sin ( 7 x )}{14}) + C

\frac{d y}{d x} = \frac{2 y}{x} + x^{2} y^{5}

\int_{0}^{1} 4 e^{5 x} d x

\int 2 - 4 x d x

\frac{d ^{2}}{d x ^{2}} (e^{9 e^{x}})

f (x) = \frac{5}{x + 5}

What is the integral of 7e^{-7x}sin(7x) ?
The integral of 7e^{-7x}sin(7x) is 7(-(e^{-7x}cos(7x))/(14)-(e^{-7x}sin(7x))/(14))+C