Can you solve the following integrals without a calculator (in seconds)?

$$\large \begin{equation} \int_2^4 \frac{ \sqrt{x} }{ \sqrt{6-x} + \sqrt{x} } \, \mathrm{d} x \end{equation} $$ $$ \large \begin{equation} \int_0^{\frac{\pi}{2}} \frac{ \sqrt{\sin(x)} }{ \sqrt{\sin(x)} + \sqrt{\cos(x)} } \, \mathrm{d} x \end{equation} $$ $$ \large \begin{equation} \int_{\sqrt[3]{\log(3)}}^{\sqrt[3]{\log(4)}} \frac{ x^2 \sin(x^3) }{ \sin(x^3) + \sin\left( \log(12) - x^3 \right) } \, \mathrm{d} x \end{equation} $$