What is the integral of xsin(x)e^x ?

The integral of xsin(x)e^x is x(-1/2 e^xcos(x)+1/2 e^xsin(x))+1/2 e^xcos(x)+C

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integral of xsin(x)e^x

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

x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) + \frac{1}{2} e^{x} cos (x) + C

Solution steps

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

Apply Integration By Parts

: x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) - \int \frac{1}{2} (- e^{x} cos (x) + e^{x} sin (x)) d x

= x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) - \int \frac{1}{2} (- e^{x} cos (x) + e^{x} sin (x)) d x

\int \frac{1}{2} (- e^{x} cos (x) + e^{x} sin (x)) d x = - \frac{1}{2} e^{x} cos (x)

= x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) - (- \frac{1}{2} e^{x} cos (x))

Simplify

= x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) + \frac{1}{2} e^{x} cos (x)

Add a constant to the solution

= x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) + \frac{1}{2} e^{x} cos (x) + C

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What is the integral of xsin(x)e^x ?
The integral of xsin(x)e^x is x(-1/2 e^xcos(x)+1/2 e^xsin(x))+1/2 e^xcos(x)+C