How do you know when to use substituion or integration by parts?

[emlekarc]

When you have a fraction, how do you know when to use iteration by parts, or use substituion, pick a u, solve for a value of x (like x=u-2) and then plug in those values?

 

[HallsofIvy]

You try both and see which works!

 

[gopher_p]

The way you know when a particular technique of integration is better suited to a particular integral over another technique is you try both ways and decide which is better. Over time and practice you will build an intuition regarding which technique will be best for a given problem in advance.

While there are some "rules of thumb" regarding when to use some techniques, the only real way to know when to use them is by practicing. Some (most in my experience) problems open themselves up to multiple techniques, and I would strongly encourage you to try them all when you have the time (i.e. when you aren't taking a timed exam). If you approach each practice problem with a "how many different ways can this be solved" mentality (i.e. it's not really solved until you know all of the solutions, not just the one you stumbled across first), you will soon figure out which techniques work best for which kinds of problems.

 

[SteamKing]

I think you mean 'integration by parts'. 'Iteration by parts' has no meaning.

 

[Giant]

In my case the textbook had separate exercises for integration by parts and substitution and at the end there were mixed problems.
Solving those individual exercises will get you familiar with the types of problems, then you test yourself by trying to solve mixed problems.
So precise answer to your question would be practice practice practice.
You should be able to find as many techniques as possible to solve one question but use the most convenient one as your final solution.

 

[sudhirking]

how do you know when to use iteration by parts, or use substituion

You ask a question which can be answered, but if we answer it for you it robs you the joy of discovering that yourself! So as others have said it is best recommended to practice and find when one technique is more suited than another. It helps not only to solve an ensemble of questions, but to take a higher position and invent questions! You will try to tinker and totter and tailor the problem around one technique! It is this vantage point that will give you your insight.

If you insist though, it is our duty to reply, no matter the hesitation it might cause. In short, the practice goes like this:

(1) always "look" for substitutions: find quantities which if exchanged with some other variable, has its derivative contained in the integrand.

For example, if I am integrating an x/(x^2-1), it is immediately advantageous to recast the denominator as a variable u since its derivative is, modulo some constant, the rest of the integrand itself!

(2) If above fails, resort to your next simple techniques (when you learn them, they might be trigonometric substitutions).

(3) If all of the above fails, it is here you concede to integration by parts, as usually it is the most exceeding in effort to calculate. Think of the method as a reverse product rule. Re-write the integrand as a derivative of the product and watch the magic happen.

For example, If I am integrating x cos(x), I might rewrite that as the derivative of the product x sin(x) subtracted from the term I was otherwise not entitled too (that is, I will additionally minus away the sin(x). I than notice that integrating the d/dx (x sin(x)) - sin(x) is fantastically easier than when in the original form.