How to find the antiderivative of e x 1 + e 2 x ?

Question

How to find the antiderivative of                               e          x                            1        +                  e                      2            x                              ?

Abbigail Joyce · Accepted Answer

write                     e      x              dx as               d          (              e        x            )                     , then we obtain            ∫                  d                  (                      e            x                    )                            1        +                              (                          e              x                        )                    2                                with the substitution   y =                     e      x              , we get            ∫                  d                  (          y          )                            1        +                  y          2                                which is equal to                  arctan              (        y        )              +    C              Now substitute back                y    =          e      x        :              arctan              (                  e          x                )              +    C

Audrina Donaldson · Accepted Answer

We need to find       ∫                  e        x                    1        +                  e                      2            x                                      d        x    =    ∫          1              1        +                              (                          e              x                        )                    2                            e      x              d        x  Now let       u    =          e      x       and so taking the differential on both sides gives       d    u    =          e      x              d      x      . Now we substitute both of these equations into the integral to get      ∫          1              1        +                  u          2                            d        u  This is a standard integral which evaluates to             arctan      u      . Substituting back for       x   we get a final answer:                              arctan          e                x            +              c

How to find the antiderivative of e x 1 + e 2 x ?

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