How to integrate 1 ( 1 + x 2 ) 2 ?

Question

How to integrate             1                                    (            1            +                          x              2                        )                    2                    ?

Campbell Davenport · Accepted Answer

I    =    ∫                  d        x                              (          1          +                      x            2                    )                2            The substitution will be used       x    =          tan      θ      , implying that             d      x        =                  sec        2            θ        d    θ  :      I    =    ∫                                        sec            2                    θ                d        θ                              (          1          +                                    tan              2                        θ                    )                2            Note that       1    +                  tan        2            θ        =                  sec        2            θ      :      I    =    ∫                                        sec            2                    θ                d        θ                              sec          4                θ              =    ∫                  d        θ                              sec          2                θ              =    ∫                  cos        2            θ        d    θ  Recall that             cos      2        θ    =    2                  cos        2            θ        -    1  , so                     cos        2            θ        =          1      2              cos      2        θ    +          1      2      .      I    =          1      2        ∫          cos      2        θ    d    θ    +    ∫          1      2        d    θ  With substitution, the first integral can be found. (try       u    =    2    θ  ).      I    =          1      4              sin      2        θ    +          1      2        θ    +    C  From       x    =          tan      θ       we see that       θ    =          arctan      x      .Furthermore, we see that             1      4              sin      2        θ    =          1      4              (      2              sin        θ                    cos        θ            )        =          1      2              sin      θ              cos      θ      .Also, since             tan      θ        =    x  , we can draw a right triangle with the side opposite       θ   being       x  , the adjacent side being       1  , and the hypotenuse being                     1        +                  x          2                    . Thus,             sin      θ        =          x                        1          +                      x            2                               and             cos      θ        =          1                        1          +                      x            2                              :      I    =          1      2              sin      θ              cos      θ        +          1      2              arctan      x        +    C        I    =          1      2              (              x                              1            +                          x              2                                          )              (              1                              1            +                          x              2                                          )        +                  arctan        x            2        +    C        I    =          x              2                  (          1          +                      x            2                    )                      +                  arctan        x            2        +    C

How to integrate 1/((1+x^2)^2)?

Answered question

Answer & Explanation

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