What is the integral of x^2(9-x^2) ?

The integral of x^2(9-x^2) is 3x^3-(x^5)/5+C

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integral of x^2(9-x^2)

\int x^{2} (9 - x^{2}) d x

3 x^{3} - \frac{x ^{5}}{5} + C

Solution steps

\int x^{2} (9 - x^{2}) d x

Expand

x^{2} (9 - x^{2}) : 9 x^{2} - x^{4}

= \int 9 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 9 x^{2} d x - \int x^{4} d x

\int 9 x^{2} d x = 3 x^{3}

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

= 3 x^{3} - \frac{x ^{5}}{5}

Add a constant to the solution

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

What is the integral of x^2(9-x^2) ?
The integral of x^2(9-x^2) is 3x^3-(x^5)/5+C