What is the integral of (1+3ln(x))/x ?

The integral of (1+3ln(x))/x is ln|x|+3/2 ln^2(x)+C

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integral of (1+3ln(x))/x

\int \frac{1 + 3 ln ( x )}{x} d x

ln ∣ x ∣ + \frac{3}{2} ln^{2} (x) + C

Solution steps

\int \frac{1 + 3 ln ( x )}{x} d x

Expand

\frac{1 + 3 ln ( x )}{x} : \frac{1}{x} + \frac{3 ln ( x )}{x}

= \int \frac{1}{x} + \frac{3 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{1}{x} d x + \int \frac{3 ln ( x )}{x} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + \frac{3}{2} ln^{2} (x)

Add a constant to the solution

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

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What is the integral of (1+3ln(x))/x ?
The integral of (1+3ln(x))/x is ln|x|+3/2 ln^2(x)+C