What is the integral of ((ln(x))^2)/(x[(ln(x))^3+8]) ?

The integral of ((ln(x))^2)/(x[(ln(x))^3+8]) is 1/3 ln|ln^3(x)+8|+C

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integral of ((ln(x))^2)/(x[(ln(x))^3+8])

\int \frac{( ln ( x ) ) ^{2}}{x [ ( ln ( x ) ) ^{3} + 8 ]} d x

\frac{1}{3} ln ln^{3} (x) + 8 + C

Solution steps

\int \frac{( ln ( x ) ) ^{2}}{x [ ( ln ( x ) ) ^{3} + 8 ]} d x

Apply u-substitution

: \int \frac{u ^{2}}{u ^{3} + 8} d u

= \int \frac{u ^{2}}{u ^{3} + 8} d u

Apply u-substitution

: \int \frac{1}{3 v} d v

= \int \frac{1}{3 v} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{3} ln ∣ v ∣

Substitute back

= \frac{1}{3} ln ln^{3} (x) + 8

Add a constant to the solution

= \frac{1}{3} ln ln^{3} (x) + 8 + C

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What is the integral of ((ln(x))^2)/(x[(ln(x))^3+8]) ?
The integral of ((ln(x))^2)/(x[(ln(x))^3+8]) is 1/3 ln|ln^3(x)+8|+C