Antiderivative of e x / ( 1 + 2 e x )I know the solution is ln ⁡ ( 2 e x + 1 ) 2 + C.However, the result I got was ln ⁡ ( e x + 0.5 ) 2 + C.What I did was: ∫ e x 1 + 2 e x d x = ∫ e x 2 ∗ ( 0.5 + e x ) d x = 0.5 ⋅ ∫ e x 0.5 + e x d x = 0.5 ⋅ ( ln ⁡ | 0.5 + e x | + C ) . I know there are antiderivative calculators online that show the correct method step by step, but I can't understand what I did wrong. What's the mistake?

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Antiderivative of       e    x        /    (  1  +  2      e    x    )I know the solution is                     ln        ⁡                  (          2                                    e                        x                    +          1          )                    2        +  C.However, the result I got was                     ln        ⁡                  (                                    e                        x                    +          0.5          )                    2        +  C.What I did was:                    ∫                              e            x                                1            +            2                          e              x                                      d        x                            =        ∫                              e            x                                2            ∗            (            0.5            +                          e              x                        )                          d        x                                          =        0.5        ⋅        ∫                              e            x                                0.5            +                          e              x                                      d        x                                          =        0.5        ⋅        (        ln        ⁡                  |                0.5        +                  e          x                          |                +        C        )        .            I know there are antiderivative calculators online that show the correct method step by step, but I can&#039;t understand what I did wrong. What&#039;s the mistake?

vikafa4g · Accepted Answer

Explanation:The others have answered your question. Note that one runs into this often also when dealing with trig functions. For example   ∫  s  i  n  x  c  o  s  x  d  x  =  s  i      n    2    (  x  )      /    2  +  C if one chooses to make the substitution   u  =  s  i  n  x, and   ∫  s  i  n  (  x  )  c  o  s  x  d  x  =  −  c  o      s    2    (  x  )      /    2  +  C, if one chooses   u  =  c  o  s  x. Both answers are correct, as of course their difference is a constant, since   s  i      n    2    (  x  )  +  c  o      s    2    (  x  )  =  1..

Serena Carpenter · Accepted Answer

Step 1These answers are the same. Namely,  ln  ⁡  (  2      e    x    +  1  )  =  ln  ⁡  (  2  (      e    x    +  0.5  )  )  =  ln  ⁡  (  2  )  +  ln  ⁡  (      e    x    +  0.5  )  .Step 2The added ln2 is absorbed by the   +  C, and so we see that the two answers are the same up to an additive constant. There is no error.

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