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Calculus/Integration by trig Substitution

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Expert: - 6/4/2008


Question

Trig Substitution
I integrated this problem by
trig sustitution.

Could you please review my answer exspecially when I
solved by putting the answer back into x form.

The book answer is ln [x + sqrt x^2 - 4]

I say  ln [ x/2  +  (sqrt x^2 + 4)/2 ]

WHO IS RIGHT ???  THANKS :)

Answer
The real answer is ln(x/2 + sqrt(x^2-4)/2) )+ C =
ln(x + sqrt(x^2-4)) - ln(2) + C =
ln (x + sqrt(x^2-4)) + C where this C = last C - ln(2).

There were two mistakes you made.  One was when it was said that
x-2^2 is x+4.  Note that the square does not apply to the minus sign, so it's still x-4.  That's the way you had it on your paper, but it was stated as a + sign in this letter.  The other is there should be parenthisis in the answer, ln[x+sqrt(x^2-4)].

Note that both of the numbers are over 2, so they can be combined on top with a 2 down below.  Since we are taking the ln of this number, we can then say that the problem is the ln of the top minus the ln of the bottom.  The bottom [ln(2)] is a constant, so it disappears as part of +C.

What it all comes down to is you and the book got the same answer and there were a couple of simple arithmetic errors.

    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    CommentThank you so much. You made this so clear.


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