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

The integral of (e^{-x}sin(x)) is -(e^{-x}cos(x))/2-(e^{-x}sin(x))/2+C

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

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

- \frac{e ^{- x} cos ( x )}{2} - \frac{e ^{- x} sin ( x )}{2} + C

Solution steps

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= - \frac{e ^{- x} cos ( x )}{2} - \frac{e ^{- x} sin ( x )}{2}

Add a constant to the solution

= - \frac{e ^{- x} cos ( x )}{2} - \frac{e ^{- x} sin ( x )}{2} + C

\int \frac{x ^{3} + 3 x ^{2}}{x ^{2} + 1} d x

(x^{2} + 1) \cdot \frac{d y}{d x} + 3 x y = 6 x

y = (1 + x) cos (x), at x = 0

\int \frac{cos ( x )}{sin ^{2} ( x ) sin ( x )} d x

\int \frac{x - 6}{x ^{2} - 18 x + 82} d x

What is the integral of (e^{-x}sin(x)) ?
The integral of (e^{-x}sin(x)) is -(e^{-x}cos(x))/2-(e^{-x}sin(x))/2+C