What is the integral of x(x-2)^3 ?

The integral of x(x-2)^3 is (x^5)/5-(3x^4)/2+4x^3-4x^2+C

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integral of x(x-2)^3

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

\frac{x ^{5}}{5} - \frac{3 x ^{4}}{2} + 4 x^{3} - 4 x^{2} + C

Solution steps

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

Expand

x (x - 2)^{3} : x^{4} - 6 x^{3} + 12 x^{2} - 8 x

= \int x^{4} - 6 x^{3} + 12 x^{2} - 8 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^{4} d x - \int 6 x^{3} d x + \int 12 x^{2} d x - \int 8 x d x

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

\int 6 x^{3} d x = \frac{3 x ^{4}}{2}

\int 12 x^{2} d x = 4 x^{3}

\int 8 x d x = 4 x^{2}

= \frac{x ^{5}}{5} - \frac{3 x ^{4}}{2} + 4 x^{3} - 4 x^{2}

Add a constant to the solution

= \frac{x ^{5}}{5} - \frac{3 x ^{4}}{2} + 4 x^{3} - 4 x^{2} + C

f (x) = 3 + 4 x^{2} - 2 x^{3}, (2, 3)

\int (1 - x^{2})^{3} d x

x \to 0 lim (\frac{9 + x}{x ^{3}})

\frac{\partial}{\partial x} ((2 x^{2} - 5 x)^{3} (2 x + 1))

What is the integral of x(x-2)^3 ?
The integral of x(x-2)^3 is (x^5)/5-(3x^4)/2+4x^3-4x^2+C