What is the integral of xcos(x)sin^2(x) ?

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

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integral of xcos(x)sin^2(x)

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

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

Solution steps

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

Apply Integration By Parts

: \frac{1}{3} x sin^{3} (x) - \int \frac{1}{3} sin^{3} (x) d x

= \frac{1}{3} x sin^{3} (x) - \int \frac{1}{3} sin^{3} (x) d x

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

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

Add a constant to the solution

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

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