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

The integral of ((x^3-x)^2)/x is 1/6 x^6-1/2 x^4+1/2 x^2+C

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integral of ((x^3-x)^2)/x

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

\frac{1}{6} x^{6} - \frac{1}{2} x^{4} + \frac{1}{2} x^{2} + C

Solution steps

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

Apply Integration By Parts

= \frac{1}{105} x^{2} (15 x^{4} - 42 x^{2} + 35) - \int - \frac{1}{105} x (15 x^{4} - 42 x^{2} + 35) d x

\int - \frac{1}{105} x (15 x^{4} - 42 x^{2} + 35) d x = - \frac{1}{210} (5 x^{6} - 21 x^{4} + 35 x^{2})

= \frac{1}{105} x^{2} (15 x^{4} - 42 x^{2} + 35) - (- \frac{1}{210} (5 x^{6} - 21 x^{4} + 35 x^{2}))

Simplify

\frac{1}{105} x^{2} (15 x^{4} - 42 x^{2} + 35) - (- \frac{1}{210} (5 x^{6} - 21 x^{4} + 35 x^{2})) : \frac{1}{6} x^{6} - \frac{1}{2} x^{4} + \frac{1}{2} x^{2}

= \frac{1}{6} x^{6} - \frac{1}{2} x^{4} + \frac{1}{2} x^{2}

Add a constant to the solution

= \frac{1}{6} x^{6} - \frac{1}{2} x^{4} + \frac{1}{2} x^{2} + C

What is the integral of ((x^3-x)^2)/x ?
The integral of ((x^3-x)^2)/x is 1/6 x^6-1/2 x^4+1/2 x^2+C