Question

Integration of dx/X+√x

Answer 1

Dear Student,
$$\begin{gathered} \int \frac{1}{x+\sqrt{x}}dx \\ lett=\sqrt{x}and{x}^{}=t^2 \\ dt=\frac{{\displaystyle 1}}{2\sqrt{x}}dx \\ dt=\frac{{\displaystyle 1}}{2t}dx \\ dx=2tdt \\ \int \frac{{\displaystyle 1}}{t^2+t}.2tdt \\ =2\int \frac{{\displaystyle t}}{t(t+1)}dt \\ =2\int \frac{{\displaystyle 1}}{(t+1)}dt \\ =2\log |t+1|+C \\ =2\log |\sqrt{x}+1|+C \end{gathered}$$
Regards