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

The integral of x^2sin(2x) is -1/2 x^2cos(2x)+1/4 (2xsin(2x)+cos(2x))+C

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integral of x^2sin(2x)

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

- \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - x cos (2 x) d x = - \frac{1}{4} (2 x sin (2 x) + cos (2 x))

= - \frac{1}{2} x^{2} cos (2 x) - (- \frac{1}{4} (2 x sin (2 x) + cos (2 x)))

Simplify

= - \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x))

Add a constant to the solution

= - \frac{1}{2} x^{2} cos (2 x) + \frac{1}{4} (2 x sin (2 x) + cos (2 x)) + C

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{6}})

x \to 7 lim (\frac{5 x}{x + 2})

t \to - 2 lim (t^{2} - 6 t - 16)

y^{^{''}} - 2 y^{^{'}} + y = e^{t} arctan (t)

\int \frac{s}{2 s + 3} d s

What is the integral of x^2sin(2x) ?
The integral of x^2sin(2x) is -1/2 x^2cos(2x)+1/4 (2xsin(2x)+cos(2x))+C