What is the integral of (-xcos(x)+x) ?

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

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integral of (-xcos(x)+x)

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

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

Solution steps

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

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

\int x d x = \frac{x ^{2}}{2}

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

Simplify

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

Add a constant to the solution

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

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