The problem is the integral of ∫ − 8 x x 4 − a 4 d x

Question

The problem is the integral of  ∫                    −        8        x                              x          4                −                  a          4                        d  x

humusen6p · Accepted Answer

Noting that  I  =  ∫            −      8      x                      x                  4                    −              a                  4                      d  x  =  −  4  ∫            d              (                  x                      2                          )                                      (                      x                          2                                )                          2                    −                        (                      a                          2                                )                          2                    Let   y  =      x          2       and   b  =      a          2      , then                    I                            =        −        4        ∫                              d            y                                              y                              2                                      −                          b                              2                                                                                      =        −        4        ∫                  1                      2            b                                    (                      1                          y              −              b                                −                      1                          y              +              b                                )                d        y                                          =        −                  2          b                (        ln        ⁡                  |                y        −        b                  |                −        ln        ⁡                  |                y        +        b                  |                )        +        C                                          =        −                  2                      a                          2                                      ln        ⁡                  |                                                    x                                  2                                            −                              a                                  2                                                                                    x                                  2                                            +                              a                                  2                                                              |                +        C                                          =                  2                      a                          2                                      ln        ⁡                  |                                                    x                                  2                                            +                              a                                  2                                                                                    x                                  2                                            −                              a                                  2                                                              |                +        C

The problem is the integral of ∫ <mrow class="MJX-TeXAtom-ORD">

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