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

The integral of x^3cos(kx) is 1/k x^3sin(kx)-3/k (-1/k x^2cos(kx)+2/(k^3)(kxsin(kx)+cos(kx)))+C

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integral of x^3cos(kx)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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