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

The integral from-c to c of (c-x^2)-(x^2-c) is -(4c^3)/3+4c^2

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integral from-c to c of (c-x^2)-(x^2-c)

\int_{- c}^{c} (c - x^{2}) - (x^{2} - c) d x

- \frac{4 c ^{3}}{3} + 4 c^{2}

Solution steps

\int_{- c}^{c} c - x^{2} - (x^{2} - c) 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_{- c}^{c} c d x - \int_{- c}^{c} x^{2} d x - \int_{- c}^{c} x^{2} - c d x

\int_{- c}^{c} c d x = 2 c^{2}

\int_{- c}^{c} x^{2} d x = \frac{2 c ^{3}}{3}

\int_{- c}^{c} x^{2} - c d x = \frac{2 c ^{3}}{3} - 2 c^{2}

= 2 c^{2} - \frac{2 c ^{3}}{3} - (\frac{2 c ^{3}}{3} - 2 c^{2})

Simplify

2 c^{2} - \frac{2 c ^{3}}{3} - (\frac{2 c ^{3}}{3} - 2 c^{2}) : - \frac{4 c ^{3}}{3} + 4 c^{2}

= - \frac{4 c ^{3}}{3} + 4 c^{2}

What is the integral from-c to c of (c-x^2)-(x^2-c) ?
The integral from-c to c of (c-x^2)-(x^2-c) is -(4c^3)/3+4c^2