Question

Question about Integration by parts.

Answer 1

So for the formula \[uv-\int\limits_{?}^{?}vdu\] how do uk what element of the integration "u" and "dv" should be cuz when im integrating, i should the wrong element, and the integration keeps repeating.

Answer 2

I choose the wrong element*

Answer 3

The answer to this is very simple. What IBP does is that it takes your integral, and reduces it to another integral where one function has been integrated whereas the other has been differentiated.

Naturally, you would take \(dv\) as the function that is easier to integrate.

Answer 4

ppl us the rule ranking types of functions. But I think it's best to look at both and decide which is easiest to integrate, or which simplifies if you differentiate. But try both, like most rules there are exceptions!

Answer 5

Agreed. You may have to experiment with your choices of u and dv. Write out two or more possibilities for integration by parts, and then choose to follow the one that seems easiest.

Answer 6

In general, the order that you prefer for \(u\) is given by the ILATE rule. You may look that up. There are not many exceptions to this.

Answer 7

Often the more complicated function is assigned to "u," and then the rest of the original integrand becomes "dv."

Answer 8

"dv" would ideally be relatively easy to integrate.

Answer 9

okay so for example this: \[\int\limits_{?}^{?}\frac{ xe ^{2x} }{ (1+2x)^{2} }dx\] how can u tell which one is most easiest to integrate, the denominator or numerator?

Answer 10

Do i do trial and error

Answer 11

I would immediately move that e^(2x) out and group it with dx: e^(2x)*dx, and then integrate.

Answer 12

Again, use the ILATE-rule.

Answer 13

Trial and error does help, sometimes. Later you will have developed the ability to spot ideal u and dv expressions.

Answer 14

I've looked up ILATE and obtained the following search results:

https://www.google.com/search?sourceid=chrome-psyapi2&ion=1&espv=2&es_th=1&ie=UTF-8&q=ILATE&oq=ILATE&aqs=chrome..69i57j0l5.2940j1j7

You might want to look through some of these results for further info.

Answer 15

so in this case, exponential comes first so, u=xe^(2x)?

Answer 16

Pretty much the opposite of that, actually. Haha.

Answer 17

oh \[\frac{ x }{ (1+2x)^{2} }\] comes first which is algebraic, so this should be u?

Answer 18

Again: Please experiment. Think of diff possibilities for u and for dv, and then decide for yourself which would be easiest for you to integrate / differentiate.

Tentatively: Yes, let u be the rational algebraic fraction, and let dv=e^(2x)*dx.

As before, if this proves to be too difficult to integrate, choose another u and dv.

Answer 19

Is "o in this case, exponential comes first so, u=xe^(2x)?" right or wrong? I won't say. Want you to experiment with it.

Answer 20

ok, so it just trial and error.

Answer 21

It's very calculated trial and error once you get the hang of it.

Answer 22

Thanks guys, I didnt even know about ILATE rule, thanks for telling me about.