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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int u d u

Apply the Power Rule

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

= - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + x^{- 1}

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

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

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

Add a constant to the solution

= - \frac{2}{3} (\frac{x + 1}{x})^{\frac{3}{2}} + C

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