Question

Решить неопределенный интеграл

Answer 1

$\int { \frac{1}{- \sqrt[4]{2x+3} + \sqrt{2x+3} } } \, dx =[ \sqrt[4]{2x+3} =t; \sqrt{2x+3} =t^{2}; 2x+3=t^{4}; \\ 2x=t^{4}-3; x= \frac{t^{4}}{2} - \frac{3}{2}; dx= 2t^{3}dt] = 2\int { \frac{t^{3}}{-t+t^{2}} } \, dt=2 \int { \frac{t^{2}}{t-1} } \, dt= \\ =[t-1=u; t=u+1; t^{2}=u^{2}+2u+1;dt=du]=2 \int { \frac{u^{2}+2u+1}{u} } \, du= \\ =u^{2}+4u+2 ln|u|+C= \sqrt{2x+3} -2 \sqrt[4]{2x+3} +1+4 \sqrt[4]{2x+3} -4+ \\ +2ln| \sqrt[4]{2x+3} -1|+C= \\ =\sqrt{2x+3} +2 \sqrt[4]{2x+3}+2ln| \sqrt[4]{2x+3} -1|+C$