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Popular Calculus >

integral of x(x-3)^2

Solution

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

Solution

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

Graph

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Frequently Asked Questions (FAQ)

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