Question

∫√(a^2-x^2)dx

Answer 1

M=x√(a^2-x^2)-∫xd(√(a^2-x^2))
=x√(a^2-x^2)+∫x^2/√(a^2-x^2)dx
=x√(a^2-x^2)+∫(x^2-a^2+a^2)/√(a^2-x^2)dx
=x√(a^2-x^2)-∫√(a^2-x^2)dx+∫a^2/√(a^2-x^2)dx
=x√(a^2-x^2)-M+a^2*arcsin(x/a)+c
所以M=(x√(a^2-x^2)+a^2*arcsin(x/a)+c)/2

求采纳，不懂请追问。

Answer 2

a^2-x^2=a^2(1-(x/a)^2)
设x/a=cost tE(0,pai) sint=(1-(x/a)^2)^(1/2) sin2t=2*x/a*(1-(x/a)^2)^(1/2)
x=acost
dx=-asintdt (a^2-x^2)^1/2=|a|sint
则：原工=∫|a|sint(-asint)dt
=-a^2∫sin^2tdt
=-a^2∫(1-cos2t)/2dt
=-a^2 *(1/2t-1/4sin(2t))+C
=a^2[(sin2t) /4-t/2)+C
将t=arccos(x/a) 与 sin2t=2*x/a*(1-(x/a)^2)^(1/2)代回即可。