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

The integral of (ln(1+x))/(sqrt(x)) is 2sqrt(x)ln(1+x)-4(-arctan(sqrt(x))+sqrt(x))+C

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

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

2 x ln (1 + x) - 4 (- arctan (x) + x) + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2 x}{1 + x} d x = 4 (- arctan (x) + x)

= 2 x ln (1 + x) - 4 (- arctan (x) + x)

Add a constant to the solution

= 2 x ln (1 + x) - 4 (- arctan (x) + x) + C

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What is the integral of (ln(1+x))/(sqrt(x)) ?
The integral of (ln(1+x))/(sqrt(x)) is 2sqrt(x)ln(1+x)-4(-arctan(sqrt(x))+sqrt(x))+C