What is the integral of xsin(2x)cos(x) ?

The integral of xsin(2x)cos(x) is -2/3 xcos^3(x)+2/3 (sin(x)-1/3 sin^3(x))+C

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integral of xsin(2x)cos(x)

\int x sin (2 x) cos (x) d x

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

Solution steps

\int x sin (2 x) cos (x) d x

Apply Integration By Parts

: - \frac{2}{3} x cos^{3} (x) - \int - \frac{2}{3} cos^{3} (x) d x

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

\int - \frac{2}{3} cos^{3} (x) d x = - \frac{2}{3} (sin (x) - \frac{sin ^{3} ( x )}{3})

= - \frac{2}{3} x cos^{3} (x) - (- \frac{2}{3} (sin (x) - \frac{sin ^{3} ( x )}{3}))

Simplify

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

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

Add a constant to the solution

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

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What is the integral of xsin(2x)cos(x) ?
The integral of xsin(2x)cos(x) is -2/3 xcos^3(x)+2/3 (sin(x)-1/3 sin^3(x))+C