换元法求积分: ∫√(x∧2-y∧2)dx
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设x=ysect,则不定积分为:
I=∫√(y^2sec^2t-y^2dysect
=∫ytant*ysect*tantdt
=y^2∫sect(sec^2t-1)dt
=y^2∫sectdtant-y^2∫dt/cost
=y^2secttant-y^2∫tantsecttantdt-y^2∫dsint/(1-sin^2t)
=y^2secttant-I-(1/2)y^2ln(1+sint)/(1-sint)
则:
I=(1/2)y^2secttant-(1/4)y^2lnln(1+sint)/(1-sint)+c
再把t代入,换成x的即可。
I=∫√(y^2sec^2t-y^2dysect
=∫ytant*ysect*tantdt
=y^2∫sect(sec^2t-1)dt
=y^2∫sectdtant-y^2∫dt/cost
=y^2secttant-y^2∫tantsecttantdt-y^2∫dsint/(1-sin^2t)
=y^2secttant-I-(1/2)y^2ln(1+sint)/(1-sint)
则:
I=(1/2)y^2secttant-(1/4)y^2lnln(1+sint)/(1-sint)+c
再把t代入,换成x的即可。
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