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Popular Calculus >

integral of x(3-x)^2

Solution

\int x (3 - x)^{2} d x

Solution

\frac{9 x ^{2}}{2} - 2 x^{3} + \frac{x ^{4}}{4} + C

Solution steps

\int x (3 - x)^{2} d x

Expand

x (3 - x)^{2} : 9 x - 6 x^{2} + x^{3}

= \int 9 x - 6 x^{2} + x^{3} 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 d x - \int 6 x^{2} d x + \int x^{3} d x

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

\int 6 x^{2} d x = 2 x^{3}

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

= \frac{9 x ^{2}}{2} - 2 x^{3} + \frac{x ^{4}}{4}

Add a constant to the solution

= \frac{9 x ^{2}}{2} - 2 x^{3} + \frac{x ^{4}}{4} + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x(3-x)^2 ?
The integral of x(3-x)^2 is (9x^2)/2-2x^3+(x^4)/4+C