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Popular >

integral of x^6sin(6x)

Solution

\int x^{6} sin (6 x) d x

Solution

- \frac{1}{6} x^{6} cos (6 x) + \frac{1}{6} x^{5} sin (6 x) + \frac{5}{36} x^{4} cos (6 x) - \frac{5}{54} x^{3} sin (6 x) - \frac{5}{108} x^{2} cos (6 x) + \frac{5}{324} x sin (6 x) + \frac{5}{1944} cos (6 x) + C

Solution steps

\int x^{6} sin (6 x) d x

Apply Integration By Parts

: - \frac{1}{6} x^{6} cos (6 x) - \int - x^{5} cos (6 x) d x

= - \frac{1}{6} x^{6} cos (6 x) - \int - x^{5} cos (6 x) d x

\int - x^{5} cos (6 x) d x = - \frac{1}{6} x^{5} sin (6 x) - \frac{5}{36} x^{4} cos (6 x) + \frac{5}{54} x^{3} sin (6 x) + \frac{5}{108} x^{2} cos (6 x) - \frac{5}{324} x sin (6 x) - \frac{5}{1944} cos (6 x)

= - \frac{1}{6} x^{6} cos (6 x) - (- \frac{1}{6} x^{5} sin (6 x) - \frac{5}{36} x^{4} cos (6 x) + \frac{5}{54} x^{3} sin (6 x) + \frac{5}{108} x^{2} cos (6 x) - \frac{5}{324} x sin (6 x) - \frac{5}{1944} cos (6 x))

Simplify

= - \frac{1}{6} x^{6} cos (6 x) + \frac{1}{6} x^{5} sin (6 x) + \frac{5}{36} x^{4} cos (6 x) - \frac{5}{54} x^{3} sin (6 x) - \frac{5}{108} x^{2} cos (6 x) + \frac{5}{324} x sin (6 x) + \frac{5}{1944} cos (6 x)

Add a constant to the solution

= - \frac{1}{6} x^{6} cos (6 x) + \frac{1}{6} x^{5} sin (6 x) + \frac{5}{36} x^{4} cos (6 x) - \frac{5}{54} x^{3} sin (6 x) - \frac{5}{108} x^{2} cos (6 x) + \frac{5}{324} x sin (6 x) + \frac{5}{1944} cos (6 x) + C

Graph

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Frequently Asked Questions (FAQ)

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