What is the integral of x*cos(x)*sin(x) ?

The integral of x*cos(x)*sin(x) is 1/2 (-1/2 xcos(2x)+1/4 sin(2x))+C

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integral of xcos(x)sin(x)

\int x \cdot cos (x) \cdot sin (x) d x

\frac{1}{2} (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x)) + C

Solution steps

\int x cos (x) sin (x) d x

Rewrite using trig identities

= \int \frac{1}{2} x sin (2 x) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x sin (2 x) d x

Apply Integration By Parts

: - \frac{1}{2} x cos (2 x) - \int - \frac{1}{2} cos (2 x) d x

= \frac{1}{2} (- \frac{1}{2} x cos (2 x) - \int - \frac{1}{2} cos (2 x) d x)

\int - \frac{1}{2} cos (2 x) d x = - \frac{1}{4} sin (2 x)

= \frac{1}{2} (- \frac{1}{2} x cos (2 x) - (- \frac{1}{4} sin (2 x)))

Simplify

= \frac{1}{2} (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x)) + C

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What is the integral of x*cos(x)*sin(x) ?
The integral of x*cos(x)*sin(x) is 1/2 (-1/2 xcos(2x)+1/4 sin(2x))+C