How do I integrate x 2 9 x 2 + 15 ?

Question

How do I integrate             x      2              9              x        2            +      15      ?

crociandomh · Accepted Answer

Lets start with            3      x              15        =  tan  ⁡  uso that            9              x        2            +      15              15        =  sec  ⁡  u  .With  d  x  =            15        3        sec    2    ⁡  u  d  uwe get                    ∫                              x            2                                9                          x              2                        +            15                          d        x                            =        ∫                                            15              9                                      tan              2                        ⁡            u                                              15                        sec            ⁡            u                          ⋅                              15                    3                          sec          2                ⁡        u        d        u                                          =                  15          27                ∫                  tan          2                ⁡        u        sec        ⁡        u        d        u                                          =                  5          9                ∫        (                  sec          2                ⁡        u        −        1        )        sec        ⁡        u        d        u                                          =                  5          9                ∫        (                  sec          3                ⁡        u        −        sec        ⁡        u        )        d        u                                          =                  5          9                                      [                                                sec            ⁡            u            tan            ⁡            u                    2                +                              ln            ⁡                          |                        sec            ⁡            u            +            tan            ⁡            u                          |                                2                −        ln        ⁡                  |                sec        ⁡        u        +        tan        ⁡        u                  |                                      ]                          +        C                                          =                  5          18                                      [                          sec        ⁡        u        tan        ⁡        u        −        ln        ⁡                  |                sec        ⁡        u        +        tan        ⁡        u                  |                                      ]                          +        C                                          =                  5          18                                      [                                                (                          9                              x                2                            +              15                        )            3            x                    15                −        ln        ⁡                              |                                                              9                              x                2                            +              15                        +            3            x                                15                                                |                                                ]                          +        C

How do I integrate x 2 </msup> <msqrt> 9

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