(1-x^2)^3开根号dx

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(1-x^2)^3开根号dx

(1-x^2)^3开根号dx
(1-x^2)^3开根号dx

(1-x^2)^3开根号dx
令x=sinu,则:u=arcsinx、cosu=√[1-(sinu)^2=√(1-x^2),dx=cosudu.
∴∫√[(1-x^2)^3]dx
=∫cosu√{[1-(sinu)^2]^3}du
=∫cosu·(cosu)^3du
=∫(cosu)^4du
=(1/4)∫[2(cosu)^2]^2du
=(1/4)∫(1+cos2x)^2du
=(1/4)∫[1+2cos2x+(cos2x)^2]du
=(1/4)u+(1/4)∫cos2xd(2x)+(1/8)∫(1+cos4x)du
=(1/4)arcsinx+(1/4)sin2u+(1/8)u+(1/32)∫cos4ud(4u)
=(3/8)arcsinx+(1/2)sinucosu+(1/32)sin4u+C
=(3/8)arcsinx+(1/2)x√(1-x^2)+(1/16)sin2ucos2u+C
=(3/8)arcsinx+(1/2)x√(1-x^2)+(1/8)sinucosu[1-2(sinu)^2]+C
=(3/8)arcsinx+(1/2)x√(1-x^2)+(1/8)x(1-2x^2)√(1-x^2)+C.