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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