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Popular >

integral of x(2-x)^2

Solution

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

Solution

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

Solution steps

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

Expand

x (2 - x)^{2} : 4 x - 4 x^{2} + x^{3}

= \int 4 x - 4 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 4 x d x - \int 4 x^{2} d x + \int x^{3} d x

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

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

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

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

Add a constant to the solution

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

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Frequently Asked Questions (FAQ)

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