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Consider the integral int_(-2)^(2) (dx)/(4 + x^(2)) . Itis easy to conclude that it is equal to (pi)/(4) .Indeed int_(-2)^(2) (dx)/(4 + x^(2)) = (1)/(2) "arc tan "(x)/(2) |_(-2)^(2)= (1)/(2) [(pi)/(4) - (-(pi)/(4))]= (pi)/(4)On the other hand, making the substitution x = (1)/(t) we havedx = - (dt)/(t^(2))|{:(x," "t),(-2,-1//2),(2,1//2):}| int_(-2)^(2) (dx)/( 4 + x^(2)) = - int_(-1//2)^(1//2) (dt)/(t^(2)(4 + (1)/(t^(2))))= - int _(-1//2)^(+1//2) (dt)/(4t^(2) + 1)=(1)/(2) "arc tan 2t |_((1)/(2))^((1)/(2)) = - (pi)/(4) This result is obviously wrong, since the integrand (1)/(4 + x^(2)) gt0 , andconsequently, thedefinite integral of this function cannot be equal to a negative number -(pi)/(4) . Find the mistake . |
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