What is the integral of ((ln(x))^2+1)/x ?

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

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integral of ((ln(x))^2+1)/x

\int \frac{( ln ( x ) ) ^{2} + 1}{x} d x

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

Solution steps

\int \frac{( ln ( x ) ) ^{2} + 1}{x} d x

Simplify

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

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{ln ^{3} ( x )}{3} + ln ∣ x ∣

Add a constant to the solution

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

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