What is the integral of (2(1-ln(x))^3)/x ?

The integral of (2(1-ln(x))^3)/x is 2ln|x|-3ln^2(x)+2ln^3(x)-1/2 ln^4(x)+C

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integral of (2(1-ln(x))^3)/x

\int \frac{2 ( 1 - ln ( x ) ) ^{3}}{x} d x

2 ln ∣ x ∣ - 3 ln^{2} (x) + 2 ln^{3} (x) - \frac{1}{2} ln^{4} (x) + C

Solution steps

\int \frac{2 ( 1 - ln ( x ) ) ^{3}}{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{( 1 - ln ( x ) ) ^{3}}{x} d x

Expand

\frac{( 1 - ln ( x ) ) ^{3}}{x} : \frac{1}{x} - \frac{3 ln ( x )}{x} + \frac{3 ln ^{2} ( x )}{x} - \frac{ln ^{3} ( x )}{x}

= 2 \cdot \int \frac{1}{x} - \frac{3 ln ( x )}{x} + \frac{3 ln ^{2} ( x )}{x} - \frac{ln ^{3} ( 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

= 2 (\int \frac{1}{x} d x - \int \frac{3 ln ( x )}{x} d x + \int \frac{3 ln ^{2} ( x )}{x} d x - \int \frac{ln ^{3} ( 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)

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

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

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

Simplify

2 (ln ∣ x ∣ - \frac{3}{2} ln^{2} (x) + ln^{3} (x) - \frac{ln ^{4} ( x )}{4}) : 2 ln ∣ x ∣ - 3 ln^{2} (x) + 2 ln^{3} (x) - \frac{1}{2} ln^{4} (x)

= 2 ln ∣ x ∣ - 3 ln^{2} (x) + 2 ln^{3} (x) - \frac{1}{2} ln^{4} (x)

Add a constant to the solution

= 2 ln ∣ x ∣ - 3 ln^{2} (x) + 2 ln^{3} (x) - \frac{1}{2} ln^{4} (x) + C

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What is the integral of (2(1-ln(x))^3)/x ?
The integral of (2(1-ln(x))^3)/x is 2ln|x|-3ln^2(x)+2ln^3(x)-1/2 ln^4(x)+C