What is the integral of (2-x^2)sin(2x+1) ?

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

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integral of (2-x^2)sin(2x+1)

\int (2 - x^{2}) sin (2 x + 1) d x

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

Solution steps

\int (2 - x^{2}) sin (2 x + 1) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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