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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

Add a constant to the solution

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

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