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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int 2 ln (u) d u

= \int 2 ln (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= 2 (u ln (u) - \int 1 d u)

\int 1 d u = u

= 2 (u ln (u) - u)

Substitute back

u = 1 + x

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

Expand

(1 + x) ln (1 + x) - (1 + x) : ln (1 + x) + x ln (1 + x) - 1 - x

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

Add a constant to the solution

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

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