What is the integral of e^{-3x}sin(x) ?

The integral of e^{-3x}sin(x) is -(e^{-3x}cos(x))/(10)-(3e^{-3x}sin(x))/(10)+C

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integral of e^{-3x}sin(x)

\int e^{- 3 x} sin (x) d x

- \frac{e ^{- 3 x} cos ( x )}{10} - \frac{3 e ^{- 3 x} sin ( x )}{10} + C

Solution steps

\int e^{- 3 x} sin (x) d x

Apply Integration By Parts

: - e^{- 3 x} cos (x) - \int 3 e^{- 3 x} cos (x) d x

= - e^{- 3 x} cos (x) - \int 3 e^{- 3 x} cos (x) d x

Take the constant out:

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

= - e^{- 3 x} cos (x) - 3 \cdot \int e^{- 3 x} cos (x) d x

Apply Integration By Parts

: e^{- 3 x} sin (x) - \int - 3 e^{- 3 x} sin (x) d x

= - e^{- 3 x} cos (x) - 3 (e^{- 3 x} sin (x) - \int - 3 e^{- 3 x} sin (x) d x)

Take the constant out:

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

= - e^{- 3 x} cos (x) - 3 (e^{- 3 x} sin (x) - (- 3 \cdot \int e^{- 3 x} sin (x) d x))

Therefore

\int e^{- 3 x} sin (x) d x = - e^{- 3 x} cos (x) - 3 (e^{- 3 x} sin (x) - (- 3 \cdot \int e^{- 3 x} sin (x) d x))

Isolate

\int e^{- 3 x} sin (x) d x

= - \frac{e ^{- 3 x} cos ( x )}{10} - \frac{3 e ^{- 3 x} sin ( x )}{10}

Add a constant to the solution

= - \frac{e ^{- 3 x} cos ( x )}{10} - \frac{3 e ^{- 3 x} sin ( x )}{10} + C

\int_{- 2}^{1} \frac{1}{x + 3} d x

x \to 0 + lim (\frac{x ^{2}}{7} - \frac{2}{x})

x \to 0 lim ((1 + 3 x)^{\frac{2}{3}})

\int (x^{2} + 2 x + 3) cos (2 x) d x

\int_{- \infty}^{0} 5^{x} d x

What is the integral of e^{-3x}sin(x) ?
The integral of e^{-3x}sin(x) is -(e^{-3x}cos(x))/(10)-(3e^{-3x}sin(x))/(10)+C