lim(a趋于0)∫(a到a+1)dx/(1+x^2+a^2)

=lim(a趋于0) (1/(1+a^2))·∫(a到a+1) dx/(1+(x/√(1+a^2))²)
=lim(a趋于0) (1/√(1+a^2))·∫(a到a+1) d(x/√(1+a^2)) /(1+(x/√(1+a^2))²)
=lim(a趋于0) (1/√(1+a^2))·arctan(x/√(1+a^2)) |(a到a+1)
=lim(a趋于0) (1/√(1+a^2))·(arctan((a+1)/√(1+a^2)) - arctan(a/√(1+a^2)))
=lim(a趋于0) (1/√(1+a^2))· lim(a趋于0) (arctan((a+1)/√(1+a^2)) - arctan(a/√(1+a^2)))
=1· lim(a趋于0) (arctan((a+1)/√(1+a^2)) - arctan(a/√(1+a^2)))
=arctan1 - arctan0
=π/4