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

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

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integral of ln(x+sqrt(x))

\int ln (x + x) d x

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

Solution steps

\int ln (x + x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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