Sunday, 20 September 2015

calculus - Show that intinfty1frac11+x3dx=int10fracx1+x3dx



Consider the integral J=011+x3dxshow that 111+x3dx=10x1+x3dxand then deduce that J=10f(x)dx where f is a function to be determined.



I'm specifically stuck on the second part of the question. It is easy to miss but the bounds for J is 0 and and not 1 and as in the case of the first part of the question.



Answer



Let ,



x=1t , dx=1t2dt



at x=,t=0



at x=1,t=1



I=0111+1t3.dtt2=01t3dt(t3+1)t2




Changing the limits,



I=10tdt1+t3=10xdx1+x3 (Replacing t by x)



J=0dx1+x3=10dx1+x3+1dx1+x3=10dx1+x3+10xdx1+x3 (From I)



J=10(x+1)dx1+x3



Thus, f(x)=x+11+x3



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