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

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

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

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

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

Solution steps

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

Expand

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

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

\int a^{4} d x = a^{4} x

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

\int x^{4} d x = \frac{x ^{5}}{5}

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

Add a constant to the solution

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

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