Prove ∫ − a a f ( x ) d x = 0assuming f ( x ) is odd.

Question

Prove       ∫          −      a              a        f  (  x  )  d  x  =  0assuming   f  (  x  ) is odd.

Quinn Everett · Accepted Answer

Let   f be odd and (Riemann) integrable (and hence almost everywhere continuous). Make the change of variable   u  =  −  x. Then:                              ∫                      −            a                    a                f        (        x        )                 d        x                            =                  ∫                      a                                −            a                          −        f        (        −        u        )                 d        u                                          =                  ∫                      a                                −            a                          f        (        u        )                 d        u                                          =        −                  ∫                      −            a                                a                          f        (        u        )                 d        u            The only real number that is equal to its negative is zero. Hence the integral is zero.The key: No use of antiderivatives needed here. Just change variables and use some intuition.

Prove <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD"> −<!-

Answered question

Answer & Explanation

New Questions in Integral Calculus