What Is Integration by Recognition and How Is It Used?

[adoado]

Hello all,

I have recently started a chapter on "Integration by recognition". To be honest, I am totally confused and I do not know why or when its used. I was hoping some more knowledgeable people would be able to clear a few things up for me ^^

• What is integration by recognition, a definition?
• What does it achieve?

I understand it has something to do with finding a derivative and then changing an integral, but how/what or why is where I am lost..

Thanks for reading,
Adrian ;)

 

[Cvan]

I've never heard of such a method explicitly. I assume it's more of your book's own terminology that simply wants you to 'recognize' how to integrate something (by substitution, by parts, by partial fractions, ...). The chapter sounds misleading.

It's not uncommon to try to recognize how a nice function would change if you take it's derivative, than look at the original function and generalize a set of rules for antiderivatives..but that only works for so many functions.
Ex) Look at x^3. We know we can find it's derivative, 3x^2. So we can make a rule for those similar integer saying that integral(x^n wrt x) = (x^(n+1))/(n+1) + some constant. Of course you have to recognize the limitations (what happens when you look at -1)!

 

[HallsofIvy]

I think you mean just a simple application of the definition of "anti-derivative": recognizing that the given function is the derivative of another:

Integrate 2x. "Oh, I recognize that! It's the derivative of x² so its integral (anti-derivative) is x²+ C." (Remembering that the derivative of any constant, C, is 0.)

Integrate cos(x). "Aha! I recognize that! I remember that the derivative of sin(x) is cos(x) so the integral of cos(x) is sin(x)+ C."

A little harder example: integrate sin(x). "Well, I almost recognize that! I remember that the derivative of cos(x) is -sin(x) so the derivative of -cos(x) is -(-sin(x))= sin(x). The integral of sin(x) is -cos(x)+ C."

Or: integrate x. "I remember that the derivative of x² is 2x so the derivative of (1/2)x² is (1/2)(2x)= x. The integral of x is (1/2)x²."