Question

Evaluate: $\int \frac{dx}{x^3(x^3+1{)}^{1/3}}$

• A. $-\frac{1}{2}{(1+\frac{1}{x^3})}^{2/3}+c$
• B. $\frac{1}{2}{(1-\frac{1}{x^3})}^{1/3}+c$
• C. $\frac{1}{2}{(1-\frac{1}{x^3})}^{2/3}+c$
• D. $-\frac{1}{2}{(1+\frac{1}{x^3})}^{1/3}+c$

Answer 1

The correct option is A $-\frac{1}{2}{(1+\frac{1}{x^3})}^{2/3}+c$

$I=\int \frac{dx}{x^3(x^3+1{)}^{1/3}}=\int \frac{dx}{x^4\sqrt[3]{3\frac{1}{x^3}+1}}$
Substitute $\frac{1}{x^3}+1=t\Rightarrow -\frac{3}{x^4}dx=dt$
$I=-\frac{1}{3}\int \frac{1}{\sqrt[3]{3t}}dt=-\frac{t^{2/3}}{2}=-\frac{1}{2}{(\frac{1}{x^3}+1)}^{2/3}$