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

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

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integral of ((x^4+2x^2-1)/(sqrt(x)))

\int (\frac{x ^{4} + 2 x ^{2} - 1}{x}) d x

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

Solution steps

\int \frac{x ^{4} + 2 x ^{2} - 1}{x} d x

Expand

\frac{x ^{4} + 2 x ^{2} - 1}{x} : x^{\frac{7}{2}} + 2 x^{\frac{3}{2}} - \frac{1}{x}

= \int x^{\frac{7}{2}} + 2 x^{\frac{3}{2}} - \frac{1}{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^{\frac{7}{2}} d x + \int 2 x^{\frac{3}{2}} d x - \int \frac{1}{x} d x

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

\int \frac{1}{x} d x = 2 x

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

Add a constant to the solution

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

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