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

The integral of |x|-(x^2-2) is 1/2 x|x|-(x^3)/3+2x+C

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integral of |x|-(x^2-2)

\int ∣ x ∣ - (x^{2} - 2) d x

\frac{1}{2} x ∣ x ∣ - \frac{x ^{3}}{3} + 2 x + C

Solution steps

\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} - 2 d x

\int ∣ x ∣ d x = \frac{x ∣ x ∣}{2}

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

= \frac{x ∣ x ∣}{2} - (\frac{x ^{3}}{3} - 2 x)

Simplify

\frac{x ∣ x ∣}{2} - (\frac{x ^{3}}{3} - 2 x) : \frac{1}{2} x ∣ x ∣ - \frac{x ^{3}}{3} + 2 x

= \frac{1}{2} x ∣ x ∣ - \frac{x ^{3}}{3} + 2 x

Add a constant to the solution

= \frac{1}{2} x ∣ x ∣ - \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 1/2 x|x|-(x^3)/3+2x+C