Compute the indefinite integral ∫ e n x e n x + 1 d x : n ∈ Z ∗

Question

Compute the indefinite integral      ∫                            e                      n            x                                                e                          n              x                                +          1                      d    x    :    n    ∈                  Z        ∗

luthersavage6lm · Accepted Answer

Set   y  =      e                            n          x                2            . Then the integral becomes  ∫                    y                              y                          2                                +          1                                2      n                          d        y            y        =            2      n        ln  ⁡  (  y  +            y              2              +    1    )  .Setting   y  =      e                            n          x                2             we obtain  I  =            2      n        ln  ⁡  (            e              n        x              +    1    +      e                            n          x                2              )  .

Jaiden Elliott · Accepted Answer

Let   u  =                    e                  n          x                                      e                      n            x                          +        1              ,  I  =  ∫  u  ⋅      2    n        (          1      u        +          u              1        −                  u          2                      )    d  u  =      2    n    ∫      1          1      −              u        2              d  u  =      1    n    ln  ⁡      |                  1        +        u                    1        −        u              |    +  CPut substitution back and simplify,  I  =      1    n    ln  ⁡      |                                        e                          n              x                                +          1                +                              e                          n              x                                                                        e                          n              x                                +          1                −                              e                          n              x                                            |    +  C  =      2    n    ln  ⁡      (                  e                  n          x                    +      1        +                  e                  n          x                      )    +  C

Compute the indefinite integral int sqrt((e^(nx))/(e^(nx)+1))dx:n in ZZ^(**)

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