Integration Methods: How Can You Determine the Best Approach?

[ada15]

Hi,

There are different ways by which we can integrate function :
1. Substitution
2. Integration by parts
3. Trigonomtery
4. Partial functions

But how one know that which method to use ?

Please can anyone explain how to figure out the way of doing integration? I'll be really thankful.

Thanks

 

[cepheid]

Hi,

Those are different integration techniques that are useful in different situations, depending upon what the form of your function being integrated is. I would recommend going back to you calculus text and looking up in what context each technique was introduced:

A summary would be:

1. Substitution: obviously useful when your integrand (the function being integrated) is actually the derivative of composition of two functions i.e. it has the form of something that has been differentiated using the chain rule.

2. By parts: useful when your integrand looks like one of the terms of something that has been differentiated using the product rule.

3. Trigonometry. This should be obvious! If your integrand has trigonometric functions, then it is applicable, otherwise it isn't.

4. Partial fractions: Umm...again fairly self explanatory. If I recall correctly, this is useful if your integrand can be decomposed using partial fractions.

 

[D H]

Integration is an art. Take integration by parts, for example. How you decide to split an expression into u and dv makes all the difference in the world. One choice makes the problem easy to solve while other choices result in a more complex integral than the original problem.

The method to use is the one that works on the problem at hand. There are some general heuristics, but they remain heuristics.

I can usually tell whether someone has integrated some equation by hand or used a program such as Maple and Mathematica to do the job for them. The programs apply the heuristics and come up with a page-long equation. Done nicely by hand, the same integral is expressed one or two lines of math.

So how to tell which technique is right? Practice.

 

[ada15]

confused:
http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/usubdirectory/USubstitution.html

 

[ada15]

http://http://dl.uncw.edu/digilib/mathematics/calculus/integration/freeze/images/FundTheorems3.gif


Lets say this question is given :
Now how do you know which method to use ?
Please can someone give me example and explain ... I have an exam tomorrow ... I am really really confused

Thanks

 

[cepheid]

I can't see any equations in your posts. Maybe they aren't showing up? Try writing them in plain text as a last resort.

 

[ada15]

I can't see any equations in your posts. Maybe they aren't showing up? Try writing them in plain text as a last resort.

int x cos 5x dx
int (lnx)^2 dx
int arctan 4t dt
int dx/5-3x

 

[ada15]

int x cos 5x dx
int (lnx)^2 dx
int arctan 4t dt
int dx/5-3x

now how do i know which method to use in these integrals ?

 

[cepheid]

int x cos 5x dx
int (lnx)^2 dx
int arctan 4t dt
int dx/5-3x

Hints:

int x cos 5x dx

you have a product of two functions. What technique is usually used here?

int (lnx)^2 dx

You have a composition of functions: u = g(x) = lnx, f(u) = u^2, and we have int f(g(x)) dx. What did I just do there? What rule applies?

int arctan 4t dt

I think this is a tricky one in which you have to express this as a product and then use a previous result involving derivative of arctan as well as by parts integration.


int dx / 5 - 3x

you have integral of "1 over something". What can you do that can reduce this to a previous problem like that?

That's about all the help I can give you.

 

[ada15]

thanks for the help.

 

[dextercioby]

For $\int \ln^{2}x \ dx$ i'd suggest part integration.

It's needed 2 times.

Daniel.

 

[AlephZero]

Basically, integration is a "box of magic tricks" - you listed 4 of the common ones. Unlike differentiation, it's not a logical process where you can follow a set of "rules" and guarantee to get the answer.

The only way to find out which trick to use for a given integral is by solving lots of problems. Try one trick, and if it doesn't seem to help then try another one.

It it's any consolation, when doing "real work" people who need to USE integrals don't bother to solve these types of problems themselves, they just look up the answers in a book or a website like http://eqworld.ipmnet.ru/en/auxiliary/aux-integrals.htm

You can use sites like that as a learning tool - look at the results (the simpler ones anyway!) and think how you could prove them using the integration methods you know. But DON'T try to learn all the answers!

 

[arildno]

You have the wrong primary focus on this.
Your primary focus when trying to integrate something should be:
Can I transform the integrand so that the result from the transformation is something I know how to integrate?

As long as the transformation itself is a permissible mathematical action, everything is allowed to be tried out.

 

[HallsofIvy]

Knowing how to integrate requires understanding each method and thinking about the individual integral. There are no "magic" formulas that can be applied without thinking!

 

[suspenc3]

I used to have the same problem, the only thing you can do is practice as many problems as you can and eventually you'll just recognise what method to use.