Integration by substitution for antiderivativesIntegration by substitution is an integration technique for definite integrals based upon this formula: ∫ a b f ( ϕ ( x ) ) ϕ ′ ( x ) d x = ∫ ϕ ( a ) ϕ ( b ) f ( u ) d u I don't understand how it can be applied in the case of indefinite integrals, how it can be used to find antiderivatives, as in this example: ∫ d x x 2 + a 2 = 1 a ∫ d u 1 + u 2 = 1 a tan − 1 ⁡ ( u ) + C = 1 a tan − 1 ⁡ ( x a ) + C The formula ∫ a b f ( ϕ ( x ) ) ϕ ′ ( x ) d x = ∫ ϕ ( a ) ϕ ( b ) f ( u ) d u is for definite integrals. Why can it be applied to indefinite integrals?

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Integration by substitution for antiderivativesIntegration by substitution is an integration technique for definite integrals based upon this formula:            ∫              a                    b              f    (    ϕ    (    x    )    )          ϕ      ′        (    x    )        d    x    =          ∫              ϕ        (        a        )                    ϕ        (        b        )                    f      (      u      )            d      u      I don&#039;t understand how it can be applied in the case of indefinite integrals, how it can be used to find antiderivatives, as in this example:      ∫                            d          x                                      x            2                    +                      a            2                                =          1      a        ∫                            d          u                          1          +                      u            2                                =          1      a              tan              −        1              ⁡    (    u    )    +    C    =          1      a              tan              −        1              ⁡          (              x        a            )        +    C  The formula             ∫              a                    b                    f      (      ϕ      (      x      )      )              ϕ        ′            (      x      )            d      x        =          ∫              ϕ        (        a        )                    ϕ        (        b        )                    f      (      u      )            d      u       is for definite integrals. Why can it be applied to indefinite integrals?

rotgelb7kjxw · Accepted Answer

Step 1Let F(u) be an antiderivative of f(u). We show that   F  (  ϕ  (  x  )  ) is an antiderivative of   f  (  ϕ  (  x  )  )      ϕ    ′    (  x  ).The proof uses the Chain Rule. Differentiate   F  (  ϕ  (  x  )  ). We get       ϕ    ′    (  x  )      F    ′    (  ϕ  (  x  )  ), which is   f  (  ϕ  (  x  )  )      ϕ    ′    (  x  ).In your particular example, we want   ∫            d      x                      x        2            +              a        2            , where   a  ≠  0. We can rewrite this as   ∫      1          a      2                  d      x              1      +      (      x              /            a              )        2            . Let   ϕ  (  x  )  =  x      /    a. Then       ϕ    ′    (  x  )  =  1      /    a. So we want   ∫      1    a        1          1      +      (      ϕ      (      x      )              )        2                  ϕ    ′    (  x  )    d  x  ..Step 2This is   ∫      1    a    ∫            d      u              1      +              u        2             where   u  =  ϕ  (  x  ). We get       1    a    arctan  ⁡  (  x      /    a  )  +  C.One does not go through all of this writing when actually using substitution. Here is a medium length version of the same thing. Let   x  =  a  u. Then   d  x  =  a    d  u and       a    2    +      x    2    =      a    2    (  1  +      u    2    ). Thus our integral is   ∫      1          a      2            1          1      +              u        2              a    d  u  ,, which is       1    a    arctan  ⁡  u  +  C. Now replace u by x/a.

Govindennz34j · Accepted Answer

Explanation:If you know that       ∫          a              b            f    (    ϕ    (    x    )    )          ϕ                                  ′                      (    x    )    d    x    =      ∫          ϕ      (      a      )              ϕ      (      b      )            f    (    u    )    d    u   then Fundamental Theorem of Calculus tells you that this can also be used for antiderivatives. Indeed, an antiderivative of   f  (  ϕ  (  x  )  )      ϕ                          ′              (  x  ) is       ∫          a              x            f    (    ϕ    (    t    )    )          ϕ                                  ′                      (    t    )    d    t    =      ∫          ϕ      (      a      )              ϕ      (      x      )            f    (    u    )    d    u      ..

Integration by substitution for antiderivatives Integration by substitution is an integration techn

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