What is the integral of (ln^3(x)-2)/x ?

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

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integral of (ln^3(x)-2)/x

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

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

Solution steps

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

Expand

\frac{ln ^{3} ( x ) - 2}{x} : \frac{ln ^{3} ( x )}{x} - \frac{2}{x}

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

Add a constant to the solution

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

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