dx/(x+根号下(1-x平方))的不定积分
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dx/(x+根号下(1-x平方))的不定积分
dx/(x+根号下(1-x平方))的不定积分
dx/(x+根号下(1-x平方))的不定积分
令x=siny,则:√(1-x^2)=√[1-(siny)^2]=cosy, y=arcsinx, dx=cosydy.
原式=∫[cosy/(siny+cosy)]dy
=∫{cosy(cosy-siny)/[(cosy)^2-(siny)^2]}dy
=∫[(cosy)^2/cos2y]dy-∫(sinycosy/cos2y)dy
=(1/2)∫[(1+cos2y)/cos2y]dy-(1/2)∫(sin2y/cos2y)dy
=(1/4)∫(1/cos2y)d(2y)+(1/2)∫dy-(1/4)∫(sin2y/cos2y)d(2y)
=(1/2)y+(1/4)∫[cos2y/(cos2y)^2]d(2y)+(1/4)∫(1/cos2y)d(cos2y)
=(1/2)arcsinx+(1/4)∫{1/[1-(sin2y)^2]}d(sin2y)+(1/4)ln|cos2y|
=(1/2)arcsinx+(1/4)ln|1-2(siny)^2|
+(1/4)∫{1/[(1+sin2y)(1-sin2y)]}d(sin2y)
=(1/2)arcsinx+(1/4)ln|1-2x^2|
+(1/8)∫[1/(1+sin2y)]d(sin2y)+(1/8)∫[1/(1-sin2y)]d(sin2y)
=(1/2)arcsinx+(1/4)ln|1-2x^2|+(1/8)∫[1/(1+sin2y)]d(1+sin2y)
-(1/8)∫[1/(1-sin2y)]d(1-sin2y)
=(1/2)arcsinx+(1/4)ln|1-2x^2|+(1/8)ln|1+sin2y|
-(1/8)ln|1-sin2y|+C
=(1/2)arcsinx+(1/4)ln|1-2x^2|+(1/8)ln|1+2sinycosy|
-(1/8)ln|1-2sinycosy|+C
=(1/2)arcsinx+(1/4)ln|1-2x^2|+(1/8)ln|1+2x√(1-x^2)|
-(1/8)ln|1-2x√(1-x^2)|+C
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