∫dx/(sin²x+tan²x)
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∫dx/(sin²x+tan²x)
∫dx/(sin²x+tan²x)
∫dx/(sin²x+tan²x)
符号难敲
∫dx/(sin²x+tan²x)
=∫[cos²x/(sin²xcos²+sin²x)]dx
=∫[(1+cos2x)/2]/[(1/4)sin²2x+(1-cos2x)/2]dx
=(1/2)∫[(1+cos2x)]/[(1/4)(1-cos²2x+(1-cos2x)/2]dx
=-2∫[(1+cos2x)]/[(cos²2x+2cos2x-3]dx
=(1/2)∫[1/(cost+3)]dt+(1/2)∫[1/(cost-1)]dt
对于∫[1/(cosx+3)]dx这类积分,
万能代换t=tan(x/2),则x=2arctant,dx=2dt/(1+t^2),cosx=(1-t^2)/(1+t^2),所以
∫dx/(cosx+3)=∫dt/(t^2+2)=1/√2×arctan(t/√2)+C=1/√2×arctan(tan(x/2)/√2)+C