What is the integral of (x+1/x)^{3/2}((x^2-1)/(x^2)) ?

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

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integral of (x+1/x)^{3/2}((x^2-1)/(x^2))

\int (x + \frac{1}{x})^{\frac{3}{2}} (\frac{x ^{2} - 1}{x ^{2}}) d x

\frac{2}{5} (\frac{x ^{2} + 1}{x})^{\frac{5}{2}} + C

Solution steps

\int (x + \frac{1}{x})^{\frac{3}{2}} \frac{x ^{2} - 1}{x ^{2}} d x

Apply u-substitution

: \int u^{\frac{3}{2}} d u

= \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = x + \frac{1}{x}

= \frac{2}{5} (x + \frac{1}{x})^{\frac{5}{2}}

Join

x + \frac{1}{x} : \frac{x ^{2} + 1}{x}

= \frac{2}{5} (\frac{x ^{2} + 1}{x})^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} (\frac{x ^{2} + 1}{x})^{\frac{5}{2}} + C

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