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∫(0→π/4)sec^2*x/(1+tanx)^2dx
题目详情
∫(0→π/4)sec^2*x/(1+tanx)^2dx
▼优质解答
答案和解析
∫sec²x/(1+tanx)²dx
=∫1/(1+tanx)²d(tanx)
=∫1/(1+tanx)²d(1+tanx)
=-1/(1+tanx)+C
∴∫(0→π/4)sec²x/(1+tanx)²dx
=[-1/(1+tanx)]|(0→π/4)
=1/(1+tan0)-1/(1+tanπ/4)
=1-1/2
=1/2
=∫1/(1+tanx)²d(tanx)
=∫1/(1+tanx)²d(1+tanx)
=-1/(1+tanx)+C
∴∫(0→π/4)sec²x/(1+tanx)²dx
=[-1/(1+tanx)]|(0→π/4)
=1/(1+tan0)-1/(1+tanπ/4)
=1-1/2
=1/2
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