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

The integral of ((x^2-1)^2)/(sqrt(x)) is 2/9 x^4sqrt(x)-4/5 x^2sqrt(x)+2sqrt(x)+C

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integral of ((x^2-1)^2)/(sqrt(x))

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

\frac{2}{9} x^{4} x - \frac{4}{5} x^{2} x + 2 x + C

Solution steps

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

Apply Integration By Parts

: x (x^{2} - 1)^{2} - \int \frac{7 x ^{4} - 6 x ^{2} - 1}{2 x} d x

= x (x^{2} - 1)^{2} - \int \frac{7 x ^{4} - 6 x ^{2} - 1}{2 x} d x

\int \frac{7 x ^{4} - 6 x ^{2} - 1}{2 x} d x = \frac{1}{45} x (35 x^{4} - 54 x^{2} - 45)

= x (x^{2} - 1)^{2} - \frac{1}{45} x (35 x^{4} - 54 x^{2} - 45)

Simplify

x (x^{2} - 1)^{2} - \frac{1}{45} x (35 x^{4} - 54 x^{2} - 45) : \frac{2}{9} x^{4} x - \frac{4}{5} x^{2} x + 2 x

= \frac{2}{9} x^{4} x - \frac{4}{5} x^{2} x + 2 x

Add a constant to the solution

= \frac{2}{9} x^{4} x - \frac{4}{5} x^{2} x + 2 x + C

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What is the integral of ((x^2-1)^2)/(sqrt(x)) ?
The integral of ((x^2-1)^2)/(sqrt(x)) is 2/9 x^4sqrt(x)-4/5 x^2sqrt(x)+2sqrt(x)+C