令x=tant
dx=sec^2 t dt
√(x^2+1)=sect
∴原式=∫sec^2 t dt /(1+tant)*sect
=∫ sect dt/(1+tant)
=∫ dt/(sint+cost)
=√2/2 ∫dt/sin(t+π/4)
=√2/2 ∫ csc(t+π/4) d(t+π/4)
=√2/2 *ln|csc(t+π/4)-cot(t+π/4)| +C
=√2 ln√[csc(t+π/4)-cot(t+π/4)] +C
追问令x=tant
dx=sec^2 t dt
√(x^2+1)=sect这边为什么是等于sect?不是可能等于-sect吗?
追答一般取正
sect与tant同符号
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