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

The integral from-a to a of (a^2-x^2)^2 is 2a^5-(4a^5)/3+(2a^5)/5

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

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

2 a^{5} - \frac{4 a ^{5}}{3} + \frac{2 a ^{5}}{5}

Solution steps

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

Expand

(a^{2} - x^{2})^{2} : a^{4} - 2 a^{2} x^{2} + x^{4}

= \int_{- a}^{a} a^{4} - 2 a^{2} x^{2} + x^{4} 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_{- a}^{a} a^{4} d x - \int_{- a}^{a} 2 a^{2} x^{2} d x + \int_{- a}^{a} x^{4} d x

\int_{- a}^{a} a^{4} d x = 2 a^{5}

\int_{- a}^{a} 2 a^{2} x^{2} d x = \frac{4 a ^{5}}{3}

\int_{- a}^{a} x^{4} d x = \frac{2 a ^{5}}{5}

= 2 a^{5} - \frac{4 a ^{5}}{3} + \frac{2 a ^{5}}{5}

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What is the integral from-a to a of (a^2-x^2)^2 ?
The integral from-a to a of (a^2-x^2)^2 is 2a^5-(4a^5)/3+(2a^5)/5