积分根号下a^2-x^2 积分根号下（a^2- - 知识问答

积分根号下a^2-x^2 积分根号下（a^2-x^2）是根号下（a^2-x^2），有个根号

1个回答

∫√（a^2-x^2）dx
设x=asint
则dx=dasint=acostdt
a^2-x^2
=a^2-a^2sint^2
=a^2cost^2
∫√（a^2-x^2）dx
=∫acost*acostdt
=a^2∫cost^2dt
=a^2∫（cos2t+1）/2dt
=a^2/4∫(cos2t+1)d2t
=a^2/4*(sin2t+2t)
将x=asint代回
∫√（a^2-x^2）dx=x√(a^2-x^2)/2+a^2*arcsin(x/a)/2

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