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

The integral of x^3cos(3x) is 1/3 x^3sin(3x)+1/3 x^2cos(3x)-2/9 xsin(3x)-2/27 cos(3x)+C

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integral of x^3cos(3x)

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

\frac{1}{3} x^{3} sin (3 x) + \frac{1}{3} x^{2} cos (3 x) - \frac{2}{9} x sin (3 x) - \frac{2}{27} cos (3 x) + C

Solution steps

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

Apply Integration By Parts

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

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

\int x^{2} sin (3 x) d x = - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))

= \frac{1}{3} x^{3} sin (3 x) - (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x)))

Simplify

\frac{1}{3} x^{3} sin (3 x) - (- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{27} (3 x sin (3 x) + cos (3 x))) : \frac{1}{3} x^{3} sin (3 x) + \frac{1}{3} x^{2} cos (3 x) - \frac{2}{9} x sin (3 x) - \frac{2}{27} cos (3 x)

= \frac{1}{3} x^{3} sin (3 x) + \frac{1}{3} x^{2} cos (3 x) - \frac{2}{9} x sin (3 x) - \frac{2}{27} cos (3 x)

Add a constant to the solution

= \frac{1}{3} x^{3} sin (3 x) + \frac{1}{3} x^{2} cos (3 x) - \frac{2}{9} x sin (3 x) - \frac{2}{27} cos (3 x) + C

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What is the integral of x^3cos(3x) ?
The integral of x^3cos(3x) is 1/3 x^3sin(3x)+1/3 x^2cos(3x)-2/9 xsin(3x)-2/27 cos(3x)+C