5. Compute
$$
\int_{0}^{1} \frac{d x}{\sqrt{x}+\sqrt[3]{x}}
$$
Answer: $\quad 5-6 \ln 2$
Solution: Writing $x=u^{6}$ so that $d x=6 u^{5} d u$, we have
$$
\begin{aligned}
\int_{0}^{1} \frac{d x}{\sqrt{x}+\sqrt[3]{x}} & =\int_{0}^{1} \frac{6 u^{5} d u}{u^{3}+u^{2}} \\
& =6 \int_{0}^{1} \frac{u^{3} d u}{u+1} \\
& =6 \int_{0}^{1}\left(u^{2}-u+1-\frac{1}{u+1}\right) d u \\
& =6\left(\frac{u^{3}}{3}-\frac{u^{2}}{2}+u-\left.\ln |u+1|\right|_{0} ^{1}\right)=5-6 \ln (2)
\end{aligned}
$$