What is the integral of x^3cos(7x) ?

The integral of x^3cos(7x) is 1/7 x^3sin(7x)-3/7 (-1/7 x^2cos(7x)+2/343 (7xsin(7x)+cos(7x)))+C

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integral of x^3cos(7x)

\int x^{3} cos (7 x) d x

\frac{1}{7} x^{3} sin (7 x) - \frac{3}{7} (- \frac{1}{7} x^{2} cos (7 x) + \frac{2}{343} (7 x sin (7 x) + cos (7 x))) + C

Solution steps

\int x^{3} cos (7 x) d x

Apply Integration By Parts

: \frac{1}{7} x^{3} sin (7 x) - \int \frac{3}{7} x^{2} sin (7 x) d x

= \frac{1}{7} x^{3} sin (7 x) - \int \frac{3}{7} x^{2} sin (7 x) d x

\int \frac{3}{7} x^{2} sin (7 x) d x = \frac{3}{7} (- \frac{1}{7} x^{2} cos (7 x) + \frac{2}{343} (7 x sin (7 x) + cos (7 x)))

= \frac{1}{7} x^{3} sin (7 x) - \frac{3}{7} (- \frac{1}{7} x^{2} cos (7 x) + \frac{2}{343} (7 x sin (7 x) + cos (7 x)))

Add a constant to the solution

= \frac{1}{7} x^{3} sin (7 x) - \frac{3}{7} (- \frac{1}{7} x^{2} cos (7 x) + \frac{2}{343} (7 x sin (7 x) + cos (7 x))) + C

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What is the integral of x^3cos(7x) ?
The integral of x^3cos(7x) is 1/7 x^3sin(7x)-3/7 (-1/7 x^2cos(7x)+2/343 (7xsin(7x)+cos(7x)))+C