求1/(x^2+1)^(3/2)的不定积分

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答案和解析

Let x = tanθ and dx = sec²θ dθ
∫ dx/(x²+1)^(3/2)
= ∫ (sec²θ)/(tan²θ+1)^(3/2) dθ
= ∫ (sec²θ)/(sec²θ)^(3/2) dθ
= ∫ (sec²θ)/(sec³θ) dθ
= ∫ cosθ dθ
= sinθ + C
= x/√(1+x²) + C

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