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Popular >

integral of x^2cos^5(2x)

Solution

\int x^{2} cos^{5} (2 x) d x

Solution

\frac{x ^{2} sin ( 2 x )}{2} + \frac{- x ^{2} sin ^{3} ( 2 x ) - 4 x cos ^{4} ( x )}{3} + \frac{x ^{2} sin ^{5} ( 2 x )}{10} + \frac{3 x cos ( 2 x )}{5} + \frac{8 x cos ^{6} ( x ) + 2 x}{9} - \frac{x cos ^{3} ( 2 x )}{15} + \frac{x cos ^{5} ( 2 x )}{50} - \frac{187}{450} sin (2 x) - \frac{1}{8} sin (4 x) - \frac{4}{27} cos^{5} (x) sin (x) + \frac{31}{27} cos^{3} (x) sin (x) - \frac{1}{225} sin^{3} (2 x) - \frac{1}{500} sin^{5} (2 x) + C

Solution steps

\int x^{2} cos^{5} (2 x) d x

\int x^{2} cos^{5} (2 x) d x = \frac{1}{2} x^{2} sin (2 x) - \frac{1}{3} x^{2} sin^{3} (2 x) + \frac{1}{10} x^{2} sin^{5} (2 x) + \frac{3}{5} x cos (2 x) - \frac{4}{3} x cos^{4} (x) + \frac{8}{9} x cos^{6} (x) + \frac{2}{9} x - \frac{1}{15} x cos^{3} (2 x) + \frac{1}{50} x cos^{5} (2 x) - \frac{187}{450} sin (2 x) - \frac{1}{8} sin (4 x) - \frac{4}{27} cos^{5} (x) sin (x) + \frac{31}{27} cos^{3} (x) sin (x) - \frac{1}{225} sin^{3} (2 x) - \frac{1}{500} sin^{5} (2 x) + C

= \frac{1}{2} x^{2} sin (2 x) - \frac{1}{3} x^{2} sin^{3} (2 x) + \frac{1}{10} x^{2} sin^{5} (2 x) + \frac{3}{5} x cos (2 x) - \frac{4}{3} x cos^{4} (x) + \frac{8}{9} x cos^{6} (x) + \frac{2}{9} x - \frac{1}{15} x cos^{3} (2 x) + \frac{1}{50} x cos^{5} (2 x) - \frac{187}{450} sin (2 x) - \frac{1}{8} sin (4 x) - \frac{4}{27} cos^{5} (x) sin (x) + \frac{31}{27} cos^{3} (x) sin (x) - \frac{1}{225} sin^{3} (2 x) - \frac{1}{500} sin^{5} (2 x) + C

Simplify

\frac{1}{2} x^{2} sin (2 x) - \frac{1}{3} x^{2} sin^{3} (2 x) + \frac{1}{10} x^{2} sin^{5} (2 x) + \frac{3}{5} x cos (2 x) - \frac{4}{3} x cos^{4} (x) + \frac{8}{9} x cos^{6} (x) + \frac{2}{9} x - \frac{1}{15} x cos^{3} (2 x) + \frac{1}{50} x cos^{5} (2 x) - \frac{187}{450} sin (2 x) - \frac{1}{8} sin (4 x) - \frac{4}{27} cos^{5} (x) sin (x) + \frac{31}{27} cos^{3} (x) sin (x) - \frac{1}{225} sin^{3} (2 x) - \frac{1}{500} sin^{5} (2 x) + C : \frac{x ^{2} sin ( 2 x )}{2} + \frac{- x ^{2} sin ^{3} ( 2 x ) - 4 x cos ^{4} ( x )}{3} + \frac{x ^{2} sin ^{5} ( 2 x )}{10} + \frac{3 x cos ( 2 x )}{5} + \frac{8 x cos ^{6} ( x ) + 2 x}{9} - \frac{x cos ^{3} ( 2 x )}{15} + \frac{x cos ^{5} ( 2 x )}{50} - \frac{187}{450} sin (2 x) - \frac{1}{8} sin (4 x) - \frac{4}{27} cos^{5} (x) sin (x) + \frac{31}{27} cos^{3} (x) sin (x) - \frac{1}{225} sin^{3} (2 x) - \frac{1}{500} sin^{5} (2 x) + C

= \frac{x ^{2} sin ( 2 x )}{2} + \frac{- x ^{2} sin ^{3} ( 2 x ) - 4 x cos ^{4} ( x )}{3} + \frac{x ^{2} sin ^{5} ( 2 x )}{10} + \frac{3 x cos ( 2 x )}{5} + \frac{8 x cos ^{6} ( x ) + 2 x}{9} - \frac{x cos ^{3} ( 2 x )}{15} + \frac{x cos ^{5} ( 2 x )}{50} - \frac{187}{450} sin (2 x) - \frac{1}{8} sin (4 x) - \frac{4}{27} cos^{5} (x) sin (x) + \frac{31}{27} cos^{3} (x) sin (x) - \frac{1}{225} sin^{3} (2 x) - \frac{1}{500} sin^{5} (2 x) + C

Graph

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