1/(1-x^2)^3/2的不定积分是怎么求的啊,急,还有x^3/(1+x^2)3/2的不定积分

Question

1/(1-x^2)^3/2的不定积分是怎么求的啊,急,还有x^3/(1+x^2)3/2的不定积分

Answer 1

∫dx/√(1-x^2)^3
=∫dx/[x^3√(1/x^2-1)^3
=(-1/2)∫d(1/x^2)/√(1/x^2-1)^3
=1/√(1/x^2-1)+C
∫x^3dx/(1+x^2)^(3/2)
=(1/2)∫x^2d(x^2)/(1+x^2)^(3/2)
=(1/2)∫dx^2/(1+x^2)^(1/2)-(1/2)∫dx^2/(1+x^2)^(3/2)
=√(1+x^2)+1/√(1+x^2)+C