What is the integral of ((1+12ln(x))^3)/x ?

The integral of ((1+12ln(x))^3)/x is (1+12ln(x))^3-1584ln^3(x)-414ln^2(x)+432ln^4(x)-35ln|x|+C

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integral of ((1+12ln(x))^3)/x

\int \frac{( 1 + 12 ln ( x ) ) ^{3}}{x} d x

(1 + 12 ln (x))^{3} - 1584 ln^{3} (x) - 414 ln^{2} (x) + 432 ln^{4} (x) - 35 ln ∣ x ∣ + C

Solution steps

\int \frac{( 1 + 12 ln ( x ) ) ^{3}}{x} d x

Apply Integration By Parts

: (1 + 12 ln (x))^{3} - \int \frac{36 ( 12 ln ( x ) + 1 ) ^{2} - ( 12 ln ( x ) + 1 ) ^{3}}{x} d x

= (1 + 12 ln (x))^{3} - \int \frac{36 ( 12 ln ( x ) + 1 ) ^{2} - ( 12 ln ( x ) + 1 ) ^{3}}{x} d x

\int \frac{36 ( 12 ln ( x ) + 1 ) ^{2} - ( 12 ln ( x ) + 1 ) ^{3}}{x} d x = 1584 ln^{3} (x) + 414 ln^{2} (x) - 432 ln^{4} (x) + 35 ln ∣ x ∣

= (1 + 12 ln (x))^{3} - (1584 ln^{3} (x) + 414 ln^{2} (x) - 432 ln^{4} (x) + 35 ln ∣ x ∣)

Simplify

= (1 + 12 ln (x))^{3} - 1584 ln^{3} (x) - 414 ln^{2} (x) + 432 ln^{4} (x) - 35 ln ∣ x ∣

Add a constant to the solution

= (1 + 12 ln (x))^{3} - 1584 ln^{3} (x) - 414 ln^{2} (x) + 432 ln^{4} (x) - 35 ln ∣ x ∣ + C

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What is the integral of ((1+12ln(x))^3)/x ?
The integral of ((1+12ln(x))^3)/x is (1+12ln(x))^3-1584ln^3(x)-414ln^2(x)+432ln^4(x)-35ln|x|+C