errors can cancel each other out
If one uses the http://planetmath.org/ChangeOfVariableInDefiniteIntegralchange of variable
for finding the value of the definite integral
the following calculation looks appropriate and faultless:
The result is quite . Unfortunately, the calculation two errors, the effects of which cancel each other out.
The crucial error in (2) is using the substitution (1) when is discontinuous in the point
on the interval of integration. The error is however canceled out by the second error using the value for , when the right value were (the values of arctan lie only between and ; see cyclometric functions). The value belongs to a different branch of the inverse tangent function than ; parts of two distinct branches cannot together form the antiderivative which must be continuous.
Then (1) is usable, and because , we obtain
|Title||errors can cancel each other out|
|Date of creation||2013-03-22 18:59:39|
|Last modified on||2013-03-22 18:59:39|
|Last modified by||pahio (2872)|