What is the integral of 5x^2sin(9x) ?

The integral of 5x^2sin(9x) is 5(-1/9 x^2cos(9x)+2/729 (9xsin(9x)+cos(9x)))+C

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integral of 5x^2sin(9x)

\int 5 x^{2} sin (9 x) d x

5 (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x))) + C

Solution steps

\int 5 x^{2} sin (9 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int x^{2} sin (9 x) d x

Apply Integration By Parts

: - \frac{1}{9} x^{2} cos (9 x) - \int - \frac{2}{9} x cos (9 x) d x

= 5 (- \frac{1}{9} x^{2} cos (9 x) - \int - \frac{2}{9} x cos (9 x) d x)

\int - \frac{2}{9} x cos (9 x) d x = - \frac{2}{729} (9 x sin (9 x) + cos (9 x))

= 5 (- \frac{1}{9} x^{2} cos (9 x) - (- \frac{2}{729} (9 x sin (9 x) + cos (9 x))))

Simplify

= 5 (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x)))

Add a constant to the solution

= 5 (- \frac{1}{9} x^{2} cos (9 x) + \frac{2}{729} (9 x sin (9 x) + cos (9 x))) + C

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What is the integral of 5x^2sin(9x) ?
The integral of 5x^2sin(9x) is 5(-1/9 x^2cos(9x)+2/729 (9xsin(9x)+cos(9x)))+C