How to evaluate ∫ x 2 + 14 x d x?

Question

How to evaluate   ∫            x      2        +    14    x    d  x?

affwysaumlgx · Accepted Answer

We need                    cosh        2            θ        -                  sinh        2            θ        =    1              cosh      2        θ    =    2                  sinh        2            θ        +    1                      sinh        2            θ        =          1      2              (              cosh        2            θ      -      1      )                  sinh      2        θ    =    2          sinh      θ              cosh      θ            a    r    c          cosh      x        =          ln              (                                            x              2                        -            1                          +        x        )            By substituting, we perform this integral, but first we simplify.            x      2        +    14    x    =          x      2        +    14    x    +    49    -    49    =                  (        x        +        7        )            2        -    49  The substitution is      x    +    7    =    7          cosh      θ                  d      x        =    7          sinh      θ        d    θ                                            (            x            +            7            )                    2                -        49              =                  49                              cosh            2                    θ                -        49                  =    7                                        cosh            2                    θ                -        1                  =    7          sinh      θ                          sinh        2            θ        =                  cosh        2            θ        -    1    =                  (                              x            +            7                    7                )            2        -    1        =                            x          2                +        14        x        +        49        -        49            49            =                            x          2                +        14        x            49                  sinh      θ        =          1      7                                x          2                +        14        x            Then,      ∫                            x          2                +        14        x                    d      x        =    ∫    7          sinh      θ        ⋅    7          sinh      θ        d    θ        =    49    ∫                  sinh        2            θ        d    θ        =          49      2        ∫          (              cosh        2            θ      -      1      )        d    θ        =          49      2              (                                    sinh            2                    θ                2            -      θ      )            =          49      2              (              sinh        θ                    cosh        θ            -      a      r      c              cosh                  (                                    x              +              7                        7                    )                    )            =          (              49        2            ⋅              1        7            ⋅                                    x            2                    +          14          x                    ⋅                        x          +          7                7            )        -          49      2              (              ln                  (                                                                                          (                    x                    +                    7                    )                                    2                                49                            -              1                                )                    +                        x          +          7                7            )        +    C        =          1      2              (      x      +      7      )                                x          2                +        14        x              -          49      2              ln              (                  |                      1            7                                (                                                            x                  2                                +                14                x                                      +                          (              x              +              7              )                        )                    |                )              +    C

How to evaluate ∫ x 2 + 14 x d x?

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