Integrating reciprocals of functions with known antiderivativesIf ∫ f ( x ) d x is known, is there a way to directly find ∫ 1 f ( x ) d x.

Question

Integrating reciprocals of functions with known antiderivativesIf       ∫                    f  (  x  )    d  x is known, is there a way to directly find       ∫                        1          f      (      x      )          d  x.

knolsaadme · Accepted Answer

Step 1Consider the function f(x) in a power series. Let                     f        (        x        )        =                  ∑                      n            =            0                                ∞                                    a                      n                                            x                      n                               then it can be seen that                               1                      f            (            x            )                          =                  1                      a                          0                                      −                              a                          1                                            a                          0                                      2                                              x        +                                            a                              1                                            2                                      −                          a                              2                                                          a                          0                                      3                                                        x                      2                          +        ⋯            Step 2Integration of each shows                    ∫        f        (        x        )                d        x                            =                  a                      0                          x        +                              a                          1                                2                                  x                      2                          +                              a                          2                                3                                  x                      3                          +        ⋯                            ∫                              d            x                                f            (            x            )                                              =                  x                      a                          0                                      −                              a                          1                                            2                                      a                              0                                            2                                                                      x                      2                          +                                            a                              1                                            2                                      −                          a                              2                                                          3                                      a                              0                                            3                                                                      x                      3                          +        ⋯

Integrating reciprocals of functions with known antiderivatives If <msubsup> ∫<!--

Answered question

Answer & Explanation

New Questions in Calculus 1