Calculus (EASY Substitution Solution)...
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Calculus (EASY Substitution Solution)...
Well, not really, but after looking at the solution to the problem and 3
days of trying to solve it, I finally have the answer. It is actually
fairly simple once you realize that the substitution is actually using the
whole equation (including the sqrt()...normally this isn't the case which
confused me)! Here is what I mean:
(Don't forget S equals that integral sign)
S (sqrt(e^x - 3)) dx
u = sqrt(e^x - 3)
u^2 + 3 = e^x
Now take the derivative of u to find the substitution to make for dx:
2u du = e^x dx
Substituting (u^2 + 3) for e^x:
(2u)/(u^2 + 3) du = dx
Now put everything back into the integral:
S [u * (2u)/(u^2 + 3)] du
S [(2u^2)/(u^2 + 3)] du
Now comes the tricky part. There is a way to split the equation up so that
there are two integrals which can be solved. This gets a bit messy, but
once the final integral is set up, it is easy to see how to finish it up:
S [(2u^2 + 6 - 6)/(u^2 + 3)] du
Why use 6 - 6? The objective is to separate and simplify the equation. It
can't get much more simpler than this:
S [(2u^2 + 6)/(u^2 + 3)] du - S [6/(u^2 + 3)] du
(Important: Pull out the 2 to cancel equation)
2 * S [(u^2 + 3)/(u^2 + 3)] du - 6 * S [1/(u^2 + 3)] du
(SIMPLIFY!!!)
2 * S du - 6 * S [1/(u^2 + 3)] du
2 * S du - 6 * S [1/(u^2 + sqrt(3)^2)] du
There are now two integrals. The first one is basic integration. The
second is able to be integrated into an arctan. Here is how it works:
2u - (6 / sqrt(3)) * arctan(u/sqrt(3)) + C
2u - 2sqrt(3) * arctan(u/sqrt(3)) + C
Now substitute sqrt(e^x - 3) for u:
2sqrt(e^x - 3) - 2sqrt(3) * arctan(sqrt(e^x - 3)/sqrt(3)) + C
This is the hardest equation that I have come across so far. For being in
independent calculus (my school doesn't go as far as I have into the book,
so I get to work on my own), I think that I'm doing fairly well. For those
of you who are thinking I'm crazy for doing calculus on my own, well, I am.
I love math. Let's just leave it at that.
Thomas J. Hruska -- thruska@tir.com
Shining Light Productions -- "Meeting the needs of fellow programmers"
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