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

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

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integral of ((ln(x))^3-(ln(x))^{1/3})/x

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

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

Solution steps

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

Apply u-substitution

: \int u^{3} - u^{\frac{1}{3}} d u

= \int u^{3} - u^{\frac{1}{3}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{3} d u - \int u^{\frac{1}{3}} d u

\int u^{3} d u = \frac{u ^{4}}{4}

\int u^{\frac{1}{3}} d u = \frac{3}{4} u^{\frac{4}{3}}

= \frac{u ^{4}}{4} - \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = ln (x)

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

Add a constant to the solution

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

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