What is the integral of-2x^6sin(3x) ?

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integral of-2x^6sin(3x)

\int - 2 x^{6} sin (3 x) d x

- 2 (- \frac{1}{3} x^{6} cos (3 x) + 2 (\frac{1}{3} x^{5} sin (3 x) - \frac{5}{3} (- \frac{1}{3} x^{4} cos (3 x) + \frac{4}{81} (9 x^{3} sin (3 x) + 9 x^{2} cos (3 x) - 6 x sin (3 x) - 2 cos (3 x))))) + C

Solution steps

\int - 2 x^{6} sin (3 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int x^{6} sin (3 x) d x

Apply Integration By Parts

= - 2 (- \frac{1}{3} x^{6} cos (3 x) - \int - 2 x^{5} cos (3 x) d x)

\int - 2 x^{5} cos (3 x) d x = - 2 (\frac{1}{3} x^{5} sin (3 x) - \frac{5}{3} (- \frac{1}{3} x^{4} cos (3 x) + \frac{4}{81} (9 x^{3} sin (3 x) + 9 x^{2} cos (3 x) - 6 x sin (3 x) - 2 cos (3 x))))

= - 2 (- \frac{1}{3} x^{6} cos (3 x) - (- 2 (\frac{1}{3} x^{5} sin (3 x) - \frac{5}{3} (- \frac{1}{3} x^{4} cos (3 x) + \frac{4}{81} (9 x^{3} sin (3 x) + 9 x^{2} cos (3 x) - 6 x sin (3 x) - 2 cos (3 x))))))

Simplify

= - 2 (- \frac{1}{3} x^{6} cos (3 x) + 2 (\frac{1}{3} x^{5} sin (3 x) - \frac{5}{3} (- \frac{1}{3} x^{4} cos (3 x) + \frac{4}{81} (9 x^{3} sin (3 x) + 9 x^{2} cos (3 x) - 6 x sin (3 x) - 2 cos (3 x)))))

Add a constant to the solution

= - 2 (- \frac{1}{3} x^{6} cos (3 x) + 2 (\frac{1}{3} x^{5} sin (3 x) - \frac{5}{3} (- \frac{1}{3} x^{4} cos (3 x) + \frac{4}{81} (9 x^{3} sin (3 x) + 9 x^{2} cos (3 x) - 6 x sin (3 x) - 2 cos (3 x))))) + C