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

The integral of x^2sin(2x)x is -1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x))+C

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integral of x^2sin(2x)x

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

- \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)) + C

Solution steps

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

Simplify

x^{2} sin (2 x) x : x^{3} sin (2 x)

= \int x^{3} sin (2 x) d x

Apply Integration By Parts

: - \frac{1}{2} x^{3} cos (2 x) - \int - \frac{3}{2} x^{2} cos (2 x) d x

= - \frac{1}{2} x^{3} cos (2 x) - \int - \frac{3}{2} x^{2} cos (2 x) d x

\int - \frac{3}{2} x^{2} cos (2 x) d x = - \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

= - \frac{1}{2} x^{3} cos (2 x) - (- \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)))

Simplify

= - \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x))

Add a constant to the solution

= - \frac{1}{2} x^{3} cos (2 x) + \frac{3}{8} (2 x^{2} sin (2 x) + 2 x cos (2 x) - sin (2 x)) + C

\int \frac{x + 1}{x ^{2} + 7 x + 10} d x

\int 12 - 8 x + x^{2} d x

\frac{d ^{2} S}{d x ^{2}} = S

x \to - \infty lim (- 3 x^{3} - x)

(2 x - 6)^{4} (x^{2} + x + 1)^{5}

What is the integral of x^2sin(2x)x ?
The integral of x^2sin(2x)x is -1/2 x^3cos(2x)+3/8 (2x^2sin(2x)+2xcos(2x)-sin(2x))+C