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

The integral of x^2sin(ax) is -1/a x^2cos(ax)+2/(a^3)(axsin(ax)+cos(ax))+C

integral of x^2sin(ax)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

d er i v a t i v e \frac{9}{r ^{3}}

a re a y = x^{2} + 5 x, y = 3 - x^{2}

d er i v a t i v e f (x) = x + 2

t an g e n t 5 x + 11

d er i v a t i v e f (x) = x^{2} + 5 x - 2

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