Question

$\int \frac{x^{\frac{1}{2}}}{1+x^{\frac{3}{4}}}dx=$

Answer 1

$\begin{array}{c}I=\int \frac{x^{\frac{1}{2}}}{1+x^{\frac{3}{4}}}dx\\ \left[\begin{array}{c}Let\left(x^{\frac{3}{4}}+1\right)=4\\ dx=\frac{4\sqrt[4]{4x}}{3}du\end{array}\right]\\ I=\frac{4}{3}\frac{4}{3}\int \left(\frac{u-1}{u}\right)du\\ I=\frac{4}{3}\int \left(1-\frac{1}{u}\right)du\\ =\frac{4}{3}u-\frac{4}{3}\log \left|u\right|+C\\ =\frac{4}{3}\left(x^{\frac{3}{4}}+1\right)-\frac{4}{3}\log \left|1+x^{\frac{3}{4}}\right|+C\end{array}$