Question

f(x)=√(r^2-x^2)，则它的原函数F(x)=？

Answer 1

r是什么呢？是半径还是？

Answer 2

F(x) = S[r^2 - x^2]^(1/2)dx
= x(r^2 - x^2)^(1/2) + Sx^2dx/(r^2 - x^2)^(1/2)
= x(r^2 - x^2)^(1/2) + S(x^2 - r^2)dx/(r^2 - x^2)^(1/2) + r^2Sdx/(r^2 - x^2)^(1/2)
= x(r^2 - x^2)^(1/2) - S(r^2-x^2)^(1/2)dx + r^2Sd(x/r)/[ 1 - (x/r)^2]^(1/2)
= x(r^2 - x^2)^(1/2) - F(x) + r^2arcsin(x/r),

F(x) = (1/2)[x(r^2 - x^2)^(1/2) + r^2arcsin(x/r)] + C,
其中，C 为任意常数。

【验算：】
F'(x) = (1/2){(r^2 - x^2)^(1/2) - x^2/(r^2 - x^2)^(1/2) + r^2(1/r)/[1 - (x/r)^2]^(1/2) }
= (1/2){ (r^2 - x^2)^(1/2) - x^2/(r^2 - x^2)^(1/2) + r^2/(r^2 - x^2)^(1/2) }
= (1/2) { (r^2 - x^2)^(1/2) + (r^2 - x^2)/(r^2 - x^2)^(1/2) }
= (r^2 - x^2)^(1/2)