What is the integral of ln(5+x) ?

The integral of ln(5+x) is 5ln(5+x)+xln(5+x)-5-x+C

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integral of ln(5+x)

\int ln (5 + x) d x

5 ln (5 + x) + x ln (5 + x) - 5 - x + C

Solution steps

\int ln (5 + x) d x

Apply u-substitution

: \int ln (u) d u

= \int ln (u) d u

Apply Integration By Parts

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

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

\int 1 d u = u

= u ln (u) - u

Substitute back

u = 5 + x

= (5 + x) ln (5 + x) - (5 + x)

Simplify

(5 + x) ln (5 + x) - (5 + x) : 5 ln (5 + x) + x ln (5 + x) - 5 - x

= 5 ln (5 + x) + x ln (5 + x) - 5 - x

Add a constant to the solution

= 5 ln (5 + x) + x ln (5 + x) - 5 - x + C

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What is the integral of ln(5+x) ?
The integral of ln(5+x) is 5ln(5+x)+xln(5+x)-5-x+C