Let I 1 = ∫ 0 π 4 ( 1 + tan ⁡ x ) 2 d x , I 2 = ∫ 0 1 1 ( 1 + x ) 2 ( 1 + x 2 ) d x then find I 1 I 2

Question

Let            I      1        =          ∫      0                        π          4                      (    1    +    tan    ⁡    x          )      2        d    x    ,                                      I        2            =              ∫        0                  1                            1                  (          1          +          x                      )            2                    (          1          +                      x            2                    )                    d      x      then find                     I        1                    I        2

Elias Flores · Accepted Answer

Note                              I          1                =                  ∫          0                                    π              4                                      (        1        +        tan        ⁡        x                  )          2                d        x                  =                      t            =            tan            ⁡            x                                    ∫          0                      1                                                (            1            +            t                          )              2                                            1            +                          t              2                                      d        t                  =                      t            →                                          1                −                t                                            1                +                t                                                    4                  I          2

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