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

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

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integral of 2xcos(x)-x^2sin(x)

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

x^{2} cos (x) + C

Solution steps

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

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

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

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

Simplify

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

= x^{2} cos (x)

Add a constant to the solution

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

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