What is the integral of x^3sin(x)cos(x) ?

The integral of x^3sin(x)cos(x) is 1/2 (-1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x)))+C

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integral of x^3sin(x)cos(x)

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

\frac{1}{2} (- \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{2} x^{3} 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^{3} sin (2 x) d x

Apply Integration By Parts

: - \frac{1}{2} x^{3} cos (2 x) - \int - \frac{3}{2} x^{2} cos (2 x) d x

= \frac{1}{2} (- \frac{1}{2} x^{3} cos (2 x) - \int - \frac{3}{2} x^{2} cos (2 x) d x)

\int - \frac{3}{2} x^{2} cos (2 x) d x = - \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

= \frac{1}{2} (- \frac{1}{2} x^{3} cos (2 x) - (- \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))))

Simplify

= \frac{1}{2} (- \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))) + C

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What is the integral of x^3sin(x)cos(x) ?
The integral of x^3sin(x)cos(x) is 1/2 (-1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x)))+C