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

= - 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

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

\int \frac{1}{x ^{2} - 13 x + 42} d x

\int \frac{5}{3 x - 2} d x

\int \frac{x - 2}{( x ^{2} + 8 x + 64 ) ^{2}} d x

\int (e^{6 x} + 7 e^{- x}) d x

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