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

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

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integral of-|x|^2

\int - ∣ x ∣^{2} d x

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

Solution steps

\int - ∣ x ∣^{2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int ∣ x ∣^{2} d x

Apply Integration By Parts

: x ∣ x ∣^{2} - \int 2 x^{2} d x

= - (x ∣ x ∣^{2} - \int 2 x^{2} d x)

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

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

Simplify

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

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

Add a constant to the solution

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

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