What is the integral of x*(x-1)^3 ?

The integral of x*(x-1)^3 is (x^5)/5-(3x^4)/4+x^3-(x^2)/2+C

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integral of x*(x-1)^3

integral

x \cdot (x - 1)^{3}

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

Solution steps

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

Expand

x (x - 1)^{3} : x^{4} - 3 x^{3} + 3 x^{2} - x

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

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

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

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

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

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

Add a constant to the solution

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

What is the integral of x*(x-1)^3 ?
The integral of x*(x-1)^3 is (x^5)/5-(3x^4)/4+x^3-(x^2)/2+C