Then, I think you probably would complete the square on the second integral.
I have a problem in my book where i have an integral and i have to use complete the square if necessary. I understand how to do the complete the square. However, this problem don't need it but its stumping me. I am not sure how the solutions manual got the second step. It split it up into two integrals and i cant see how it came to that conclusion?
The next step is
How are they splitting up the integral and where is the 2x +6 in the first integral and -6 in the second coming from?
The next step( which i left out) looks like they use complete the square on the second integral.
Thank you so much!!!!!
I understand your formula. However, i am still confused on how i would have known to use a 6? You said it make it easier but how would i have known that? Why couldnt i have used a 3 instead? Im sorry but im just baffled.
I understand your confusion. When you see a fraction that you have to integrate, there are a few things to try. It really is a case of "manipulate, guess and check."
-Partial fractions. Here this isn't possible because the denominator doesn't factor.
-Substitutions. Here, there *might* be a substitution that will make life easier but I certainly wouldn't spot it easily.
If neither of these will work, then check whether you can somehow make something of the form - what my teacher used to call a "related integral" for some reason.
Start with the denominator. Here, this is . If we differentiate this, we get . If this is our numerator, then great! We have something of the form . But it isn't.
is of the form but is not.
We cannot easily integrate
...so we make it into:
which we can then integrate. We don't try '3' because this does not relate to the denominator. We get the '6' from differentiating the denominator.