I have the following definite integral: 10 ∫ 0 ln ⁡ 3 e t d t e 2 t + 4

Question

I have the following definite integral:  10      ∫    0          ln      ⁡      3                          e        t            d      t                      e                  2          t                    +      4

sepolturamo · Accepted Answer

Letting   u  =      e    t    ,  d  u  =      e    t    d  t gives         10          ∫      0              ln        ⁡        3                                      e          t                d        t                              e                      2            t                          +        4              =    10          ∫      1      3              1                        u          2                +        4              d    u  Now let   u  =  2  tan  ⁡  θ  ,  d  u  =  2      sec    2    ⁡  θ to get  10            ∫              u        =        1                    u        =        3                    1              2        sec        ⁡        θ              ⋅    2          sec      2        ⁡    θ    d    θ    =    10          ∫              u        =        1                    u        =        3              sec    ⁡    θ    d    θ    =    10                  [        ln        ⁡                              |                          sec        ⁡        θ        +        tan        ⁡        θ                              |                          ]                    u        =        1                    u        =        3                  =    10                  [        ln        ⁡                              |                                                              u              2                        +            4                    2                +                  u          2                                      |                          ]            1              3              =    10                  [        ln        ⁡        (                              u            2                    +          4                +        u        )        ]            1      3        =    10          (      ln      ⁡      (              13            +      3      )      −      ln      ⁡      (              5            +      1      )      )

I have the following definite integral: 10 <msubsup> ∫ 0 <m

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