Question

$\int \sqrt{\left(\frac{x}{1+x^3}\right)}$dx is equal to $\frac{n}{m}\log \{t+\sqrt{p+t^k}\}.$FInd $k+m+p+n$ ? where $x^{3/2}=t$

Answer 1

Let $I=\int \frac{x^{\frac{1}{2}}}{\sqrt{1+x^3}}dx$
Substitue $x^{\frac{3}{2}}=t\Rightarrow \frac{3}{2}{x}^{\frac{1}{2}}dx=dt$
$\therefore I=\frac{2}{3}\int \frac{dt}{\sqrt{1+t^2}}=\frac{2}{3}\log \{t+\sqrt{1+t^2}\}\therefore k+m+p+n=2+3+1+2=8$