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

The integral of x^2(a^2-x^2)^2 is (a^4x^3)/3-(2a^2x^5)/5+(x^7)/7+C

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

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

\frac{a ^{4} x ^{3}}{3} - \frac{2 a ^{2} x ^{5}}{5} + \frac{x ^{7}}{7} + C

Solution steps

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

Expand

x^{2} (a^{2} - x^{2})^{2} : a^{4} x^{2} - 2 a^{2} x^{4} + x^{6}

= \int a^{4} x^{2} - 2 a^{2} x^{4} + x^{6} 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} x^{2} d x - \int 2 a^{2} x^{4} d x + \int x^{6} d x

\int a^{4} x^{2} d x = \frac{a ^{4} x ^{3}}{3}

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

\int x^{6} d x = \frac{x ^{7}}{7}

= \frac{a ^{4} x ^{3}}{3} - \frac{2 a ^{2} x ^{5}}{5} + \frac{x ^{7}}{7}

Add a constant to the solution

= \frac{a ^{4} x ^{3}}{3} - \frac{2 a ^{2} x ^{5}}{5} + \frac{x ^{7}}{7} + C

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What is the integral of x^2(a^2-x^2)^2 ?
The integral of x^2(a^2-x^2)^2 is (a^4x^3)/3-(2a^2x^5)/5+(x^7)/7+C