I tried solving the following integral using integral by parts : ∫ x 3 ( x 2 + 1 ) 2 d x

Question

I tried solving the following integral using integral by parts :  ∫            x      3              (              x        2            +      1              )        2              d  x

Agrebbef4l93 · Accepted Answer

Are correct since the two answers differ by a constant.                    ∫                              x                          3                                                          (                              x                                  2                                            +              1              )                                      2                                      d        x                            =        −                  1          2                ∫                  x                      2                          d                  (                      1                                          x                                  2                                            +              1                                )                                                  =        −                              x                          2                                            2                          (                              x                                  2                                            +              1              )                                      +                  1          2                ∫                              2            x                                              x                              2                                      +            1                          d        x                                          =        −                              x                          2                                            2                          (                              x                                  2                                            +              1              )                                      +                  1          2                ln        ⁡                  (                      x                          2                                +          1          )                +                  C                      1                                                            =        −                                            x                              2                                      +            1            −            1                                2                          (                              x                                  2                                            +              1              )                                      +                  1          2                ln        ⁡                  (                      x                          2                                +          1          )                +                  C                      1                                                            =        −                  1          2                +                  1                      2                          (                              x                                  2                                            +              1              )                                      +                  1          2                ln        ⁡                  (                      x                          2                                +          1          )                +                  C                      1                                                            =                  1                      2                          (                              x                                  2                                            +              1              )                                      +                  1          2                ln        ⁡                  (                      x                          2                                +          1          )                +                  C                      2                          ,            where       C          2        =      C          1        −      1    2

I tried solving the following integral using integral by parts : ∫

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