Identity and relationship between integrals and antiderivativesIf I have that G ( a b ) = ∫ 1 a b d t t = ∫ 1 a d t t + ∫ a a b d t t .. G ( a b ) = G ( a ) + ∫ a a b d t t .How is that true? Specifically this line is reasoning G ( banana ) = ∫ 1 banana d t t As clever as banana man is he forgot to explain why that is true and I can't find it in any of my books or wikipedia entries. What does it mean?Does this mean that the area under the graph of 2 from 1 to 2 is given as ∫ 1 2 1 d x?

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Identity and relationship between integrals and antiderivativesIf I have that   G  (  a  b  )  =      ∫    1          a      b                  d      t        t    =      ∫    1    a              d      t        t    +      ∫    a          a      b                  d      t        t    ..  G  (  a  b  )  =  G  (  a  )  +      ∫    a          a      b                  d      t        t    .How is that true? Specifically this line is reasoning  G  (      banana    )  =      ∫    1                  banana                        d      t        t  As clever as banana man is he forgot to explain why that is true and I can&#039;t find it in any of my books or wikipedia entries. What does it mean?Does this mean that the area under the graph of 2 from 1 to 2 is given as       ∫    1    2    1  d  x?

trajeronls · Accepted Answer

Step 1It seems that you are confused as to how they are defining G(x). That is, G(x) is defined as follows:  G  (  x  )  =      ∫    1    x        1    t      d  tThus, by definition of G, you have that:  G  (  banana  )  =      ∫    1    banana        1    t      d  tNow, for the if/then statement, we make use of the theorem that:      ∫    a    b    f  (  x  )    d  x  +      ∫    b    c    f  (  x  )    d  x  =      ∫    a    c    f  (  x  )    d  x(If it helps, G(t) is actually just the natural log, lnt.)Step 2EDIT: I&#039;m going to try to further explain this, as it appears that your question wasn&#039;t really about the if/then statement, but rather about a sort of function called an &quot;integral function.&quot;In an integral function, the variable is not in the integrand, but rather in one of the limits. That is, we can define an integral function G(x) as:  G  (  x  )  =      ∫    a    x    f  (  t  )    d  tNote that the variable of the function (x) is different than the variable of integration (t).Thus,   G  (  1  )  =      ∫    a    1    f  (  t  )    d  t.It is very important to note that this does not imply that   f  (  1  )  =      ∫    a    1    f  (  t  )    d  t.

Identity and relationship between integrals and antiderivatives If I have that G ( a b

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