Question

不定积分x^2/(a^2-x^2)^1/2 dx怎么求 求过程

Answer 1

=∫a^2(x/a)^2/(1-(x/a)^2)^0.5dx/a
x/a=sin(t)
=∫a^2sin^2(t)/cos(t)dsin(t)
=a^2∫sin^2(t)dt
=a^2∫(1/2-(1-2sin^2(t))/2)dt
= a^2∫(1/2dt-a^2∫cos(2t))/4)d2t
= a^2(t-sin(2t)/4)+C
t=arcsin(x/a)
= a^2(t-sin(2t)/4)+C
= a^2(arcsin(x/a)-sin(arcsin(x/a))cos(arcsin(x/a))/2)+C
= a^2arcsin(x/a)-x(a^2-x^2)^0.5/2+C

Answer 2

∫x^2dx/√(a^2-x^2)
=∫-xd√(a^2-x^2)
=-x√(a^2-x^2)+∫√(a^2-x^2)dx
=-x√(a^2-x^2)+(a^2/2)(arcsin(x/a)+(x/a)(√1-(x^2/a^2))+C
x=asinu ∫√(a^2-x^2)dx=∫a^2cosu^2du=(a^2/2)(u+sinucosu)+C