What is the integral of ln(x+6) ?

The integral of ln(x+6) is xln(x+6)+6ln(x+6)-x-6+C

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integral of ln(x+6)

\int ln (x + 6) d x

x ln (x + 6) + 6 ln (x + 6) - x - 6 + C

Solution steps

\int ln (x + 6) 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 = x + 6

= (x + 6) ln (x + 6) - (x + 6)

Simplify

(x + 6) ln (x + 6) - (x + 6) : x ln (x + 6) + 6 ln (x + 6) - x - 6

= x ln (x + 6) + 6 ln (x + 6) - x - 6

Add a constant to the solution

= x ln (x + 6) + 6 ln (x + 6) - x - 6 + C

\int (x + 2) e^{3 x} d x

\int e^{x} 3 e^{x} + 2 d x

\int \frac{- 7 x ^{3}}{x ^{2} + 9} d x

\int \frac{x - x}{x ^{2}} d x

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

What is the integral of ln(x+6) ?
The integral of ln(x+6) is xln(x+6)+6ln(x+6)-x-6+C