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

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

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integral of ln(sqrt(x+2))

\int ln (x + 2) d x

\frac{1}{2} x ln (x + 2) + ln (x + 2) - \frac{1}{2} x - 1 + C

Solution steps

\int ln (x + 2) d x

Simplify

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

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

Take the constant out:

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

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

Apply u-substitution

: \int ln (u) d u

= \frac{1}{2} \cdot \int ln (u) d u

Apply Integration By Parts

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

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

\int 1 d u = u

= \frac{1}{2} (u ln (u) - u)

Substitute back

u = x + 2

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

Simplify

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

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

Add a constant to the solution

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

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