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

The integral of (ln(x))/(xsqrt(2+ln(x))) is 2ln(x)sqrt(2+ln(x))-4/3 (2+ln(x))^{3/2}+C

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integral of (ln(x))/(xsqrt(2+ln(x)))

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

2 ln (x) 2 + ln (x) - \frac{4}{3} (2 + ln (x))^{\frac{3}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2 2 + ln ( x )}{x} d x = \frac{4}{3} (2 + ln (x))^{\frac{3}{2}}

= 2 ln (x) 2 + ln (x) - \frac{4}{3} (2 + ln (x))^{\frac{3}{2}}

Add a constant to the solution

= 2 ln (x) 2 + ln (x) - \frac{4}{3} (2 + ln (x))^{\frac{3}{2}} + C

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What is the integral of (ln(x))/(xsqrt(2+ln(x))) ?
The integral of (ln(x))/(xsqrt(2+ln(x))) is 2ln(x)sqrt(2+ln(x))-4/3 (2+ln(x))^{3/2}+C