What is the integral of xsin(x)+x^2cos(x) ?

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

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integral of xsin(x)+x^2cos(x)

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

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

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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

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