1) x^3*(x^2-1)^(1/3) dx
2) 1/(xln(x^3)) dx
Appreciate all help given. Thank you.
2. Let $\displaystyle u=\ln(x)$ and remember that: $\displaystyle \ln(x^3)=3\cdot{\ln(x)}$
1. Note that: $\displaystyle
x^3 \cdot \left( {x^2 - 1} \right)^{\tfrac{1}
{3}} = x \cdot \left( {x^2 - 1 + 1} \right) \cdot \left( {x^2 - 1} \right)^{\tfrac{1}
{3}} = x \cdot \left( {x^2 - 1} \right)^{\tfrac{1}
{3} + 1} + x \cdot \left( {x^2 - 1} \right)^{\tfrac{1}
{3}}
$
and now try a substitution
Hello,
$\displaystyle =x^2 \cdot x \cdot (x^2-1)^{\frac 13}$
Integrate by parts :
$\displaystyle u(x)=x^2$
$\displaystyle \begin{aligned} v'(x) &=x \cdot (x^2-1)^{\frac 13} \\
&=\frac 12 \cdot 2x \cdot (x^2-1)^{\frac 13} \end{aligned}$
$\displaystyle \Longrightarrow \begin{aligned} v(x) &=\frac 12 \cdot \frac{(x^2-1)^{\frac 43}}{\frac 43} \\
&=\frac 38 \cdot (x^2-1)^{\frac 43} \end{aligned}$
etc...