What is the integral of x(1-x)^2 ?

The integral of x(1-x)^2 is (x^2)/2-(2x^3)/3+(x^4)/4+C

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integral of x(1-x)^2

\int x (1 - x)^{2} d x

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

Solution steps

\int x (1 - x)^{2} d x

Expand

x (1 - x)^{2} : x - 2 x^{2} + x^{3}

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

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

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

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

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

Add a constant to the solution

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

x \to - 2 lim (x^{2} + 5 x + 4)

\int \frac{x + 2}{x + 1} d x

\frac{d}{d x} (2 (3 x^{2} + x - 3)^{2})

\int \frac{x ^{2} - 4}{x ^{2}} d x

x \to 1 lim (\frac{x - 2}{x ^{2} - 4})

What is the integral of x(1-x)^2 ?
The integral of x(1-x)^2 is (x^2)/2-(2x^3)/3+(x^4)/4+C