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`T h ev a l u eofint_0^oo(dx)/(1+x^4)i s``s a m ea st h a tofint_0^oo(x^2+1dx)/(1+x)``pi/(2sqrt(2))``s a m ea st h a tofint_0^oo(x^2+1dx)/(1+x^4)`(d) `pi/(sqrt(2))`A. same as that of `int_(0)^(oo) (x^(2)+1dx)/(1+x^(4))`B. `(pi)/(2sqrt(2))`C. same as that of `int_(0)^(oo) (x^(2)dx)/(1+x^(4))`D. `(pi)/(sqrt(2))` |
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Answer» Correct Answer - B::C `I=int_(0)^(oo) (dx)/(1+x^(4))` ……………1 `=int_(0)^(oo) (x^(2)+1-x^(2))/(1+x^(4))dx` `=int_(0)^(oo)(x^(2))/(1+x^(4))dx+int_(0)^(oo)(1-x^(2))/(1+x^(4))dx=I_(1)+I_(2)` `I_(2)=int_(0)^(oo) (1/(x^(2))-1)/(1/(x^(2))+x^(2))dx` Put `x+1/x=y` `:.I_(2)=int_(oo)^(oo) (-1)/(y^(2)-2)dy=0` `:.I=int_(0)^(oo) (dx)/(1+x^(4))=int_(0)^(oo)(x^(2)dx)/(1+x^(4))` Adding 1 and 2 we get `2I=int_(0)^(oo) (1+x^(2)dx)/(1+x^(4))=int_(0)^(oo) (1/(x^(2))+1)/(1/(x^(2))+x^(2))dx` (Putting `x=1/x=`) `=int_(-oo)^(oo) (dy)/(y^(2)+2)` `=[1/(sqrt(2))"tan"^(-1)y/(sqrt(2))]_(-oo)^(oo) =(pi)/(sqrt(2))` or `(pi)/(2sqrt(2))` |
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