What is the integral of ln(1+1/(sqrt(x))) ?

The integral of ln(1+1/(sqrt(x))) is xln(1+1/(sqrt(x)))+sqrt(x)+1-ln|sqrt(x)+1|+C

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integral of ln(1+1/(sqrt(x)))

\int ln (1 + \frac{1}{x}) d x

x ln (1 + \frac{1}{x}) + x + 1 - ln x + 1 + C

Solution steps

\int ln (1 + \frac{1}{x}) d x

Apply Integration By Parts

: x ln (1 + \frac{1}{x}) - \int - \frac{1}{2 ( x + 1 )} d x

= x ln (1 + \frac{1}{x}) - \int - \frac{1}{2 ( x + 1 )} d x

\int - \frac{1}{2 ( x + 1 )} d x = - x - 1 + ln x + 1

= x ln (1 + \frac{1}{x}) - (- x - 1 + ln x + 1)

Simplify

= x ln (1 + \frac{1}{x}) + x + 1 - ln x + 1

Add a constant to the solution

= x ln (1 + \frac{1}{x}) + x + 1 - ln x + 1 + C

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What is the integral of ln(1+1/(sqrt(x))) ?
The integral of ln(1+1/(sqrt(x))) is xln(1+1/(sqrt(x)))+sqrt(x)+1-ln|sqrt(x)+1|+C