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

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

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

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

- x^{2} cos (x) + 2 x sin (x) + cos (x) + C

Solution steps

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

Apply Integration By Parts

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

\int - 2 x cos (x) d x = - 2 (x sin (x) + cos (x))

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

Simplify

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

= - x^{2} cos (x) + 2 x sin (x) + cos (x)

Add a constant to the solution

= - x^{2} cos (x) + 2 x sin (x) + cos (x) + C

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