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

The integral of x^2sin(2x+5) is -1/2 x^2cos(2x+5)+1/2 xsin(2x+5)+1/4 cos(2x+5)+C

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integral of x^2sin(2x+5)

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

- \frac{1}{2} x^{2} cos (2 x + 5) + \frac{1}{2} x sin (2 x + 5) + \frac{1}{4} cos (2 x + 5) + C

Solution steps

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

Apply Integration By Parts

: - \frac{1}{2} x^{2} cos (2 x + 5) - \int - x cos (2 x + 5) d x

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

\int - x cos (2 x + 5) d x = - \frac{1}{2} x sin (2 x + 5) - \frac{1}{4} cos (2 x + 5)

= - \frac{1}{2} x^{2} cos (2 x + 5) - (- \frac{1}{2} x sin (2 x + 5) - \frac{1}{4} cos (2 x + 5))

Simplify

= - \frac{1}{2} x^{2} cos (2 x + 5) + \frac{1}{2} x sin (2 x + 5) + \frac{1}{4} cos (2 x + 5)

Add a constant to the solution

= - \frac{1}{2} x^{2} cos (2 x + 5) + \frac{1}{2} x sin (2 x + 5) + \frac{1}{4} cos (2 x + 5) + C

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What is the integral of x^2sin(2x+5) ?
The integral of x^2sin(2x+5) is -1/2 x^2cos(2x+5)+1/2 xsin(2x+5)+1/4 cos(2x+5)+C