intfrac x21x dxSolution For the integrand sqrtx21x substitut

/int/frac/ x2-1x dx

Solution

For the integrand sqrt(x^2-1)/x, substitute x = sec(u) and dx = tan(u) sec(u) du. Then sqrt(x^2-1) = sqrt(sec^2(u)-1) = tan(u) and u = sec^(-1)(x): = integral tan^2(u) du Write tan^2(u) as sec^2(u)-1: = integral (sec^2(u)-1) du Integrate the sum term by term: = integral -1 du+ integral sec^2(u) du The integral of sec^2(u) is tan(u): = tan(u)+ integral -1 du The integral of -1 is -u: = tan(u)-u+constant Substitute back for u = sec^(-1)(x): = sqrt(x^2-1)-sec^(-1)(x)+constant Which is equivalent for restricted x values to: = sqrt(x^2-1)+tan^(-1)(1/sqrt(x^2-1))+constant