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

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

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

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

2 ln 1 + x^{\frac{1}{2}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

= 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = 1 + x^{\frac{1}{2}}

= 2 ln 1 + x^{\frac{1}{2}}

Add a constant to the solution

= 2 ln 1 + x^{\frac{1}{2}} + C

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