求不定积分(x^3)*sqrt(a^2-x^2)
问题描述：
求不定积分(x^3)*sqrt(a^2-x^2)
答
设 x= asint => dx = acost dt
则原积分变为：∫(asint)^3√a²-a²sin²t × acost dt
=a^4∫(sint)^3cos²t dt
=a^4∫(sint)^3-(sint)^5 dt
∵ (sint)^3=(-1/4)sin3t+(3/4)sint
(sint)^5=(1/16)sin5t-(5/16)sin3t+(5/8)sint
∴a^4∫(sint)^3-(sint)^5 dt
＝a^4∫(-1/16)sin5t+(1/16)sin3t+(1/8)sint dt
＝a^4((-1/80)cos5t+(1/48)cos3t+(1/8)cost +C)