What is the integral of (ln(2sqrt(x)+1))/(sqrt(x)) ?

The integral of (ln(2sqrt(x)+1))/(sqrt(x)) is 2sqrt(x)ln(2sqrt(x)+1)-2sqrt(x)-1+ln|2sqrt(x)+1|+C

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integral of (ln(2sqrt(x)+1))/(sqrt(x))

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

2 x ln (2 x + 1) - 2 x - 1 + ln 2 x + 1 + C

Solution steps

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

Apply Integration By Parts

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

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

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

= 2 x ln (2 x + 1) - (2 x + 1 - ln 2 x + 1)

Simplify

= 2 x ln (2 x + 1) - 2 x - 1 + ln 2 x + 1

Add a constant to the solution

= 2 x ln (2 x + 1) - 2 x - 1 + ln 2 x + 1 + C

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