求不定积分x/(1-x)^3dx

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求不定积分x/(1-x)^3dx
求不定积分x/(1-x)^3dx

求不定积分x/(1-x)^3dx
∫x/(1-x)³dx
=-∫(1-x-1)/(1-x)³dx
=-∫(1-x)/(1-x)³dx+∫1/(1-x)³dx
=∫1/(1-x)²d(1-x)-∫1/(1-x)³d(1-x)
=-1/(1-x)+1/2(1-x)²+C

§x/(1-x)>3dx=§dx/(1-x)>3-§(1-x)dx/(1-x)>3=-§d(1-x)/(1-x)>3+§d(1-x)/(1-x)2=1/[(1-x)>2*2]-1/(1-x)=(2x-1)/2(1-x)>2

第二类换元积分法：令x=siny => dx=cosy dy cosy=√(1-x )，tany=x+ C = x /[4(1-x ) ] + C ∫x^3/(1-x^2)^3dx =1/2

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