What is the integral of (4-ln(x+3))/(x+3) ?

The integral of (4-ln(x+3))/(x+3) is 4ln|x+3|-(ln^2(x+3))/2+C

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integral of (4-ln(x+3))/(x+3)

\int \frac{4 - ln ( x + 3 )}{x + 3} d x

4 ln ∣ x + 3 ∣ - \frac{ln ^{2} ( x + 3 )}{2} + C

Solution steps

\int \frac{4 - ln ( x + 3 )}{x + 3} d x

Expand

\frac{4 - ln ( x + 3 )}{x + 3} : \frac{4}{x + 3} - \frac{ln ( x + 3 )}{x + 3}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4}{x + 3} d x - \int \frac{ln ( x + 3 )}{x + 3} d x

\int \frac{4}{x + 3} d x = 4 ln ∣ x + 3 ∣

\int \frac{ln ( x + 3 )}{x + 3} d x = \frac{ln ^{2} ( x + 3 )}{2}

= 4 ln ∣ x + 3 ∣ - \frac{ln ^{2} ( x + 3 )}{2}

Add a constant to the solution

= 4 ln ∣ x + 3 ∣ - \frac{ln ^{2} ( x + 3 )}{2} + C

What is the integral of (4-ln(x+3))/(x+3) ?
The integral of (4-ln(x+3))/(x+3) is 4ln|x+3|-(ln^2(x+3))/2+C