原式=lim(n→∞)1/n*∑(i=1→n)1/√(1+(i/n)^2)
=∫(0→1)dx/√(1+x^2)
令x=tant
则原式=∫(0→π/4)sec^2(t)dt/sect
=∫(0→π/4)dt/cost
=∫(0→π/4)costdt/cos^2(t)
=∫(0→π/4)d(sint)/(1-sin^2(t))
=1/2*∫(0→π/4)(1/(1+sint)+1/(1-sint))d(sint)
=1/2*(ln(1+sint)-ln(1-sint))|(0→π/4)
=ln(√2+1)