Question

Дифференцирование по частям.

Answer 1

$\displaystyle \int arcsin(\sqrt\frac{x}{x+1})dx=x*arcsin(\sqrt\frac{x}{x+1})-\frac{1}{2}\int\frac{\sqrt xdx}{(x+1)}=\\=x*arcsin(\sqrt\frac{x}{x+1})-\sqrt x+arctg\sqrt x+C\\\\\\\\u=arcsin(\sqrt\frac{x}{x+1})\rightarrow du=\sqrt{x+1}*(\frac{1}{(x+1)^2}*\frac{\sqrt{x+1}}{2\sqrt x})dx=\\=\frac{dx}{2\sqrt x(x+1)}\\dv=dx\rightarrow v=x\\\\\\\int\frac{\sqrt xdx}{(x+1)}=2\int\frac{t^2dt}{t^2+1}=2\int\frac{t^2+1-1}{t^2+1}dt=2\int(1-\frac{1}{t^2+1})=\\=2(t-arctgt)+C=2(\sqrt x-arctg\sqrt x)+C\\x=t^2\\dx=2tdt$