I need to find the following integral: ∫ 1 ( 1 + x ) 1 + x − x 2 d x .I tried using Euler's formula and put x t + 1 = 1 + x − x 2 and after to do integration in parts but that goes nowhere. Is there anotehr method to solve it?

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I need to find the following integral:  ∫      1          (      1      +      x      )              1        +        x        −                  x          2                      d  x  .I tried using Euler&#039;s formula and put   x  t  +  1  =      1    +    x    −          x      2       and after to do integration in parts but that goes nowhere. Is there anotehr method to solve it?

luthersavage6lm · Accepted Answer

Hint We proceed by successive variable change:  I  =  ∫      1          (      1      +      x      )              1        +        x        −                  x          2                      d  x  =      2          5        ∫            d      x              (      x      +      1      )              1        −        (                              2            x            −            1                                5                                    )          2                    with   t  =            2      x      −      1              5       we find  I  =  2  ∫            d      t              (              5            t      +      3      )              1        −                  t          2                      .Now by   t  =  sin  ⁡  (  θ  ) we have  I  =  2  ∫            d      θ                      5            sin      ⁡      θ      +      3      Finally, we pose   s  =  tan  ⁡  (  θ      /    2  ) to find the desiblack result.

Alexia Avila · Accepted Answer

I have tried few steps but i m not sure thats ok or not.  ∫      1          (      1      +      x      )              1        +        x        −                  x          2                      d  x  .let   1  +  x  =  thence   d  x  =  d  t  ∫      1          (      1      +      x      )              1        +        x        −                  x          2                      d  x  .  =  ∫      1          t              t        −        (        t        −        1                  )          2                      d  t  =  ∫      1                  t        3            −      (              t        2            )      (      t      −      1              )        2              d  t

I need to find the following integral: int((1)/((1+x)sqrt(1+x-x^2))dx. I tried using Euler's formula and put xt+1=sqrt(1+x-x^2) and after to do integration in parts but that goes nowhere. Is there anotehr method to solve it?

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