Question

Evaluate: $\int \frac{dx}{\sqrt[3]{3x^2}(1+\sqrt[3]{3x})}$ using Euler's substitution

• A. $3\ln |-1+\sqrt[3]{3x}|+C$
• B. $3\ln |1+\sqrt[3]{3x}|+C$
• C. $3\ln |-1-\sqrt[3]{3x}|+C$
• D. None of these

Answer 1

The correct option is B $3\ln |1+\sqrt[3]{3x}|+C$

We need to find value of $\int \frac{dx}{x^{\frac{2}{3}}(1+x^{\frac{1}{3}})}$

Let $z=1+x^{\frac{1}{3}}$

$dz=\frac{1}{3}{x}^{\frac{-2}{3}}dx$

$3dz=x^{\frac{-2}{3}}dx$

$I=\int \frac{3dz}{z}$

$=3\ln z+c$

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