Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. ∫ 1 1 + e x d x

Question

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.   ∫      1          1      +              e                  x                      d  x

Baluttor7 · Accepted Answer

∫      1          1      +              e                  x                      d  xApply the negative exponent rule       1          a              n              =      a          −      n        ∫      1          1      +              e                  x                      d  x  =  ∫      1          1      +              1                  e                      −            x                                d  x  =  ∫            e              −        x                            e                  −          x                    +      1        d  xLet   u  =      e          −      x        +  1  ⇒  d  u  =  −      e          −      x        d  x  ⇒  −  d  u  =      e          −      x        d  x  ∫            e              −        x                            e                  −          x                    +      1        d  x  =  ∫            −      d      u        u    =  −  ∫            d      u        u  Integrate  −  ∫            d      u        u    =  −  ln  ⁡      |    u      |    +  CPutting the answer back in terms of x,   u  =      e          −      x        +  1  −  ln  ⁡  (      e          −      x        +  1  )  +  CResult:  −  ln  ⁡  (      e          −      x        +  1  )  +  C

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. int 1/(1+e^(x))dx

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