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

The integral of x^2sin(bx) is -1/b x^2cos(bx)+2/(b^3)(bxsin(bx)+cos(bx))+C

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integral of x^2sin(bx)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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