Question

这道题的答案是？大佬求解

Answer 1

设 x = a * sinθ。那么，θ = arcsin(x/a)，dx = a * cosθ *dθ。则这个不定积分可以变换为：
=∫(a²sin²θ)*(a*cosθ*dθ)/√(a²-a²sin²θ)
=∫(a²sin²θ)*(a*cosθ*dθ)/(a*cosθ)
=∫(a²sin²θ)dθ
=a²/2 * ∫(2sin²θ)dθ
=a²/2 * ∫[1-cos(2θ)]dθ
=a²/2 * [∫dθ - ∫cos(2θ)dθ]
=a²/2 * [θ - sin(2θ)/2] + C
=a²/2 * [θ - sinθcosθ] + C
=a²/2 * θ - (a*sinθ)*(a*cosθ)/2 + C
=a²/2 * arcsin(x/a) - x/2 * √(a²-x²) + C

Answer 2

积分表里面刚好有这个，望采纳