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

The integral of x-(x^2-2) is (x^2)/2-(x^3)/3+2x+C

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integral of x-(x^2-2)

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

\frac{x ^{2}}{2} - \frac{x ^{3}}{3} + 2 x + C

Solution steps

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

Simplify

= \int x - x^{2} + 2 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 d x - \int x^{2} d x + \int 2 d x

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

\int x^{2} d x = \frac{x ^{3}}{3}

\int 2 d x = 2 x

= \frac{x ^{2}}{2} - \frac{x ^{3}}{3} + 2 x

Add a constant to the solution

= \frac{x ^{2}}{2} - \frac{x ^{3}}{3} + 2 x + C

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What is the integral of x-(x^2-2) ?
The integral of x-(x^2-2) is (x^2)/2-(x^3)/3+2x+C