What is the integral of (ln(x+4))/(2x+8) ?

The integral of (ln(x+4))/(2x+8) is 1/4 ln^2(x+4)+C

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integral of (ln(x+4))/(2x+8)

\int \frac{ln ( x + 4 )}{2 x + 8} d x

\frac{1}{4} ln^{2} (x + 4) + C

Solution steps

\int \frac{ln ( x + 4 )}{2 x + 8} d x

Factor out common term

2 : 2 (x + 4)

= \int \frac{ln ( x + 4 )}{2 ( x + 4 )} 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 \frac{ln ( x + 4 )}{x + 4} d x

Apply u-substitution

: \int \frac{ln ( u )}{u} d u

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

Apply u-substitution

: \int v d v

= \frac{1}{2} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{2} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{2} \cdot \frac{ln ^{2} ( x + 4 )}{2}

Simplify

\frac{1}{2} \cdot \frac{ln ^{2} ( x + 4 )}{2} : \frac{1}{4} ln^{2} (x + 4)

= \frac{1}{4} ln^{2} (x + 4)

Add a constant to the solution

= \frac{1}{4} ln^{2} (x + 4) + C

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What is the integral of (ln(x+4))/(2x+8) ?
The integral of (ln(x+4))/(2x+8) is 1/4 ln^2(x+4)+C