Wednesday, October 6, 2010

Oh, That Darn "+C"

We were working on integration using trigonometric substitutions today in Calculus II, and the last problem of the day was the one below.
The integral
I was chatting with the students in the last couple of minutes of class, and I showed them the TI-92 calculator emulator that I have on my laptop.  When I did the integral on the calculator, this is what I got.
Output from VTI
The integrals looked close, but there is a factor of 1/4 in the answer that we worked by hand that seemed to disappear when the calculator computed the integral.  I was stumped to see how they where the same.

When I got back to my office one minute later, I sat at my desk and the answer hit me.  The factor of 1/4 was absorbed by the arbitrary constant as follows.
Now I see!
So, the arbitrary constant comes back to get us again.  That is why math teachers are so picky about it.