∫(arctansqrt(x)/(sqrt(x)*(1+x)))dx

Question

∫(arctansqrt(x)/(sqrt(x)*(1+x)))dx

Answer 1

Замена переменной (arctg√x) = y  - >
1/((1+x)*2√x)dx = dy  т.к.   (arctgx) ' = 1/(1+x^2)
->  dx = dy*(2√x)*(1+x)   ->  ∫ (   ) dx =
= ∫ (y/((√x)*(1+x))*dy*(1+x)*2√x) = ∫ 2ydy = y^2 + C =
(arctg√x)^2 + C