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