Calculate ∫ 5 x + 1 e 2 x − 1 d x

Question

Calculate   ∫      5          x      +      1            e          2      x      −      1          d  x

getrdone07tl · Accepted Answer

Use integration by parts. Set   2  x  =  y and you&#039;ll get:  5      e          −      1        ∫      5                  y        2                  e    y    d  y  =  5      e          −      1        ∫      5                  y        2              d  (      e    y    )Then  5      e          −      1        ∫      5                  y        2                  e    y    d  y  =  5      e          −      1        ∫      5                  y        2              d  (      e    y    )  =  5      e          −      1        (      5                  y        2                  e    y    −            ln      ⁡      (      5      )        2    ∫      5                  y        2                  e    y    d  y  )Combining the first and the last parts:  5      e          −      1        ∫      5                  y        2                  e    y    d  y  =  5      e          −      1        (      5                  y        2                  e    y    −            ln      ⁡      (      5      )        2    ∫      5                  y        2                  e    y    d  y  )Cancelling out multipliers and collecting the integral terms:  (  1  +            ln      ⁡      (      5      )        2    )  ∫      5                  y        2                  e    y    d  y  =      5                  y        2                  e    y  So one shoulf get:  ∫      5                  y        2                  e    y    d  y  =                    5                              y            2                                      e        y                    1      +                        ln          ⁡          (          5          )                2            So after that just remember the multiplier   5      e          −      1       and the original variable   2  x  =  y

Kasey Reese · Accepted Answer

Another way to solve:  ∫      5          x      +      1            e          2      x      −      1        d  x  =  ∫      5    e    (  5      e    2        )    x    d  x  (  5      e    2        )    x    =  u  ⇒  (  5      e    2        )    x    d  x  =            d      u              ln      ⁡      (      5              e        2            )        ∫      5    e    (  5      e    2        )    x    d  x  =      5          e      ln      ⁡      (      5              e        2            )        ∫  d  u  =      5          e      ln      ⁡      (      5              e        2            )        (  u  +  C  )

Calculate int 5^(x+1)e^(2x-1)dx

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