What is the integral of-xe^xsin(x) ?

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

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integral of-xe^xsin(x)

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

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

Solution steps

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

Take the constant out:

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

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

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

Add a constant to the solution

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

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

f (x) = (9 x^{6} + 2 x^{3})^{4}

\int \frac{sec ( x )}{sin ( x )} d x

\frac{d}{d x} ((sin (3 x))^{3})

x \to 0 lim ((1 - \frac{1}{x}) x)

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