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

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

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integral of x(x-a)^3

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

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

Solution steps

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

Expand

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

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

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

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

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

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

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

Add a constant to the solution

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

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