On what integration domain is the author finding the anti derivative?Find the antiderivative(s): ∫ 1 x 2 − a 2 d x. ∫ 1 x 2 − a 2 d x ∫ 1 ( x + a ) ( x − a ) d x 1 2 a ∫ ( 1 x − a − 1 x + a ) d x 1 2 a [ ln ⁡ ( x − a ) − ln ⁡ ( x + a ) ] + C 1 2 a ln ⁡ ( | x − a x + a | ) + C , x ≠ ± a 1 x 2 − a 2 is a discontinuous function, and there are three integration domains of it: ( − ∞ , − a ), ( − a , a ) and ( a , + ∞ ). We will get three different family of functions as antiderivatives for the three different integration domains of this function, and two antiderivatives belonging to two different integration domains/family of functions will not differ by a constant.My question is that my book found out the antiderivative belonging to which integration domain?

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On what integration domain is the author finding the anti derivative?Find the antiderivative(s):   ∫            1                        x          2                −                  a          2                          d    x.  ∫            1                        x          2                −                  a          2                          d    x  ∫            1              (        x        +        a        )        (        x        −        a        )                  d    x      1          2      a        ∫      (                  1                  x          −          a                    −              1                  x          +          a                      )        d    x      1          2      a        [  ln  ⁡  (  x  −  a  )  −  ln  ⁡  (  x  +  a  )  ]  +  C      1          2      a        ln  ⁡  (      |              x      −      a              x      +      a            |    )  +  C  ,  x  ≠  ±  a      1                  x        2            −              a        2             is a discontinuous function, and there are three integration domains of it:   (  −  ∞  ,  −  a  ),   (  −  a  ,  a  ) and   (  a  ,  +  ∞  ). We will get three different family of functions as antiderivatives for the three different integration domains of this function, and two antiderivatives belonging to two different integration domains/family of functions will not differ by a constant.My question is that my book found out the antiderivative belonging to which integration domain?

Misael Li · Accepted Answer

Step 1If the component pieces of a function are governed by the same &#039;rule&#039; that has a primitive (e.g., this is not so in the case of   f  (  x  )  =      |        1    x        |    )  ,, then, from the Fundamental Theorem of Calculus, we do expect their primitives to differ by a constant.(It is accurate to say that on each integration interval,       1    x   has primitive   ln  ⁡      |    x      |    .  )Step 2In the given solution, there has been no additional condition imposed on the integrand (to apply a certain law/theorem, for example) on top of the starting restriction   x  ≠  ±  a  ,, so there&#039;s no basis for inferring that the result doesn&#039;t apply on   (  −  ∞  ,  −  a  ) or   (  −  a  ,  a  ) or   (  a  ,  ∞  )  .

Armeninilu · Accepted Answer

Step 1Your book used the absolute value in logarithm and this makes the antiderivative having a unique expression. It gives the same formula for the three domains.If your integrand was   f  (  x  )  =      1                  |                    x        2            −      1              |            Step 2You won&#039;t get the same expression. in each of the intervalls   (  −  ∞  ,  −  1  )    (  −  1  ,  1  )    ,  (  1  ,  ∞  ), there is a special antiderivative.

On what integration domain is the author finding the anti derivative? Find the antiderivative(s):

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