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

The integral of x^2*sin(nx) is -1/n x^2cos(nx)+2/(n^3)(nxsin(nx)+cos(nx))+C

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integral of x^2*sin(nx)

\int x^{2} \cdot sin (n x) d x

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

Solution steps

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

Apply Integration By Parts

: - \frac{1}{n} x^{2} cos (n x) - \int - \frac{2}{n} x cos (n x) d x

= - \frac{1}{n} x^{2} cos (n x) - \int - \frac{2}{n} x cos (n x) d x

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of x^2*sin(nx) ?
The integral of x^2*sin(nx) is -1/n x^2cos(nx)+2/(n^3)(nxsin(nx)+cos(nx))+C