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Popular Calculus >

integral of x^6sin(4x)

Solution

\int x^{6} sin (4 x) d x

Solution

- \frac{1}{4} x^{6} cos (4 x) + \frac{3}{2} (\frac{1}{4} x^{5} sin (4 x) - \frac{5}{512} (- 32 x^{4} cos (4 x) + 32 x^{3} sin (4 x) + 24 x^{2} cos (4 x) - 12 x sin (4 x) - 3 cos (4 x))) + C

Solution steps

\int x^{6} sin (4 x) d x

Apply Integration By Parts

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

\int - \frac{3}{2} x^{5} cos (4 x) d x = - \frac{3}{2} (\frac{1}{4} x^{5} sin (4 x) - \frac{5}{512} (- 32 x^{4} cos (4 x) + 32 x^{3} sin (4 x) + 24 x^{2} cos (4 x) - 12 x sin (4 x) - 3 cos (4 x)))

= - \frac{1}{4} x^{6} cos (4 x) - (- \frac{3}{2} (\frac{1}{4} x^{5} sin (4 x) - \frac{5}{512} (- 32 x^{4} cos (4 x) + 32 x^{3} sin (4 x) + 24 x^{2} cos (4 x) - 12 x sin (4 x) - 3 cos (4 x))))

Simplify

= - \frac{1}{4} x^{6} cos (4 x) + \frac{3}{2} (\frac{1}{4} x^{5} sin (4 x) - \frac{5}{512} (- 32 x^{4} cos (4 x) + 32 x^{3} sin (4 x) + 24 x^{2} cos (4 x) - 12 x sin (4 x) - 3 cos (4 x)))

Add a constant to the solution

= - \frac{1}{4} x^{6} cos (4 x) + \frac{3}{2} (\frac{1}{4} x^{5} sin (4 x) - \frac{5}{512} (- 32 x^{4} cos (4 x) + 32 x^{3} sin (4 x) + 24 x^{2} cos (4 x) - 12 x sin (4 x) - 3 cos (4 x))) + C

Graph

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Frequently Asked Questions (FAQ)

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