How to approch Integrals and determine how to solve

[XedLos]

I have a general question what are some techniques to identifying what type of integration you need to make when given an integral.
How do you make the choice what to choos as u or v when doing integration by parts?
What about trig integration. What are the steps you take to solve the problem?

Integration teqniques i ahve learned:
By parts
Trig integrals
Trig sub
Partial fractions
Improer integrals

 

[danago]

I reakon the best way to learn is by practicing as many as you can, and you'll start recongnizing when to use which method.

 

[rocomath]

Look at the problem first.

Normal functions: parts
Square root functions: trig sub
Rational functions: partial fractions

 

[snipez90]

When I first see an integral, substitution is a technique that comes to mind first. I'm sure you've learned "u-sub", as some textbooks like to claim that the two fundamental techniques are "u-sub" and integration by parts.

Personally, I prefer substitution because it is usually very flexible and elegant. Some textbooks will try to make "u-sub" and trig sub seem like very different techniques (sometimes giving formulas for trig sub) but they are essentially the same technique based on the chain rule. Typically if you have a square root in the denominator and that's the ONLY expression in the denominator, you want to try a trig substitution (it's easy to find the correct substitution, since it is based off of the pythagorean identities). However, that is not always the case, and it's not always a good idea to generalize trig sub to "square roots". If there are two expressions multiplied in the denominator, one with square roots, u-sub may work better. Sometimes you won't find square roots but a trig substitution would work nicely (think of the pythagorean identities).

Improper integrals are pretty easy to spot. Look for infinity in the limits of integration and keep in mind where the denominator of the integrand goes to zero.

Denominators that factor nicely lend themselves to partial fractions. There is also a quick way of determining the coefficients of the separate expressions. As you may have noticed, paying attention to the denominator of any integrand is important.

Integration by parts (IBP) is somewhat of a last resort for me. It's pretty easy to see if all other methods will fail or be inefficient. IBP is based on a simple derivation, but in many cases it's a pretty dull and inelegant method. Anyways, to determine u and dv, it helps to think about which functions are difficult to integrate or easy to differentiate and vice versa. Consider natural logarithm functions. It's hard to find the antiderivative, but very very easy to differentiate. If you see a natural logarithm, it's usually a good idea to make that your "u" (make sure you see why). Inverse trig functions are also rather annoying to integrate, but they are easy to differentiate, so that's also a good choice for "u".

Polynomials are in the middle of the spectrum. They are easy to integrate and differentiate. Use this to you advantage (u or dv, depending on what the other function is). Exponentials are very easy to differentiate, and fairly easy to integrate. Considering that a natural log or inverse trig expression may be the other function, you'll want to select the exponential function as "dv" (between a polynomial and an exponential, use your judgement). Lastly, certain trig functions are very easy to integrate, and easy to differentiate. As in the case of the exponential, you'll usually want to make this "dv" if paired with a natural log or inverse trig function. When it's a trig function and a polynomial, letting "u" be the polynomial has the advantage that you'll probably end up having to integrate an expression of similar or usually simpler terms (because the du in v*du will be of one less power than u). use your judgment for a trig function and an exponential.

 

[snipez90]

Anyways, I've written a lot on integration by parts because while it may be inelegant, it is very useful. There are ways to make it more efficient.

What I wrote on IBP was not comprehensive. It was meant to illustrate the kind of reasoning that goes towards effectively picking u and dv. Worst case scenario is you picked incorrectly the first time and have to try it the other way, not bad. But we can be lazy and fruitful.

The acronym LIPET will help you decide which function should be u and which should be dv. L stands for natural Logarithm. I stands for Inverse trig. P is Polynomial. E is Exponential. And T is Trig. The idea is that if you've narrowed it down to IBP and have two functions, select the the function that matches with the first function to appear in LIPET to be "u" and the other dv. This turns out to work pretty well, but I'll let you decide.

There is also a method of IBP known as tabular integration (perhaps informally?). It is pretty efficient but there will be cases where it won't work.

 

[JG89]

Another technique not listed is to try to rewrite the integrand. This works especially well with trig identities. For example, suppose you have to find the indefinite integral of sin(2x). Use the identity sin(2x) = 2sinxcosx. All you have to do now to integrate is factor out the 2 and use a u = sinx substitution.