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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

Simplify

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

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

Add a constant to the solution

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

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