What is the integral of (ln(x+1))/(\sqrt[3]{x+1)} ?

The integral of (ln(x+1))/(\sqrt[3]{x+1)} is 3/2 (x+1)^{2/3}ln(x+1)-9/4 (x+1)^{2/3}+C

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integral of (ln(x+1))/(\sqrt[3]{x+1)}

\frac{3}{2} (x + 1)^{\frac{2}{3}} ln (x + 1) - \frac{9}{4} (x + 1)^{\frac{2}{3}} + C

Solution steps

Apply u-substitution

Apply Integration By Parts

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

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

\int \frac{3}{2 u ^{\frac{1}{3}}} d u = \frac{9}{4} u^{\frac{2}{3}}

= \frac{3}{2} u^{\frac{2}{3}} ln (u) - \frac{9}{4} u^{\frac{2}{3}}

Substitute back

u = x + 1

= \frac{3}{2} (x + 1)^{\frac{2}{3}} ln (x + 1) - \frac{9}{4} (x + 1)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{2} (x + 1)^{\frac{2}{3}} ln (x + 1) - \frac{9}{4} (x + 1)^{\frac{2}{3}} + C

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What is the integral of (ln(x+1))/(\sqrt[3]{x+1)} ?
The integral of (ln(x+1))/(\sqrt[3]{x+1)} is 3/2 (x+1)^{2/3}ln(x+1)-9/4 (x+1)^{2/3}+C