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

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

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integral of sin(2x)x^2

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

\int_{0}^{4} 2 π x (x) d x

\int (6 x)^{3} d x

n = 1 \sum \infty 4^{n} \cdot x^{n}

n = 1 \sum \infty \frac{1}{3 n + 1}

\frac{d}{d x} (csc (x^{2}))

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