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

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

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

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

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

Solution steps

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

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

\int 2 d x = 2 x

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

Simplify

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

Add a constant to the solution

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

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