What is the integral of ((x^2)/2-x)cos(2x) ?

The integral of ((x^2)/2-x)cos(2x) is

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integral of ((x^2)/2-x)cos(2x)

\int (\frac{x ^{2}}{2} - x) cos (2 x) d x

\frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) - \frac{x ^{3}}{6} + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{2} x sin (2 x) - \frac{1}{8} sin (2 x) - \frac{1}{4} cos (2 x) + C

Solution steps

\int (\frac{x ^{2}}{2} - x) cos (2 x) d x

Rewrite using trig identities

= \int - \frac{x ^{2}}{2} + x + x^{2} cos^{2} (x) - 2 x cos^{2} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{x ^{2}}{2} d x + \int x d x + \int x^{2} cos^{2} (x) d x - \int 2 x cos^{2} (x) d x

\int \frac{x ^{2}}{2} d x = \frac{x ^{3}}{6}

\int x d x = \frac{x ^{2}}{2}

\int x^{2} cos^{2} (x) d x = \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{8} sin (2 x)

\int 2 x cos^{2} (x) d x = x^{2} + \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)

= - \frac{x ^{3}}{6} + \frac{x ^{2}}{2} + \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{8} sin (2 x) - (x^{2} + \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))

Simplify

- \frac{x ^{3}}{6} + \frac{x ^{2}}{2} + \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{8} sin (2 x) - (x^{2} + \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) : \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) - \frac{x ^{3}}{6} + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{2} x sin (2 x) - \frac{1}{8} sin (2 x) - \frac{1}{4} cos (2 x)

= \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) - \frac{x ^{3}}{6} + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{2} x sin (2 x) - \frac{1}{8} sin (2 x) - \frac{1}{4} cos (2 x)

Add a constant to the solution

= \frac{1}{3} x^{3} cos^{2} (x) - \frac{1}{6} x^{3} cos (2 x) - \frac{x ^{3}}{6} + \frac{1}{4} x^{2} sin (2 x) + \frac{1}{4} x cos (2 x) - \frac{1}{2} x sin (2 x) - \frac{1}{8} sin (2 x) - \frac{1}{4} cos (2 x) + C

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What is the integral of ((x^2)/2-x)cos(2x) ?
The integral of ((x^2)/2-x)cos(2x) is