作业帮lim(a趋于0)∫(a到a+1)dx/(1+x^2+a^2)
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/04/29 10:10:06
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
=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
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