Question Number 10147 by ridwan balatif last updated on 27/Jan/17

∫(√(p/(p−1)))dp=...?

Commented byprakash jain last updated on 27/Jan/17

p=x^2   dp=2xdx  =∫(√(x^2 /(x^2 −1))) 2xdx  =∫ ((2x^2 )/(√(x^2 −1))) dx  substitute x=cosh t  dx=sinh t dt  =∫2cosh^2 t dt  =∫(1+cosh 2t)dt  =(t+(1/2)sinh 2t)+C  =(t+sinh tcosh t)+C  =(cosh^(−1) x+x(√(x^2 −1)))+C  =ln (x+(√(x^2 −1)))+x(√(x^2 −1))+C  =ln ((√p)+(√(p−1)))+(√(p(p−1)))+C