Question

Note: This is NOT a question. This is a tutorial.

How to know what to use as "u" in integration by parts? See comment below to see how!

Answer 1

When solving integrals using integration by part, the biggest problem is which you'll use as u. You want your u to be the one that is hard to integrate. This is the list on which you'll use as u (it is arranged in descending order).

Logarithms
Inverse Trigonometric Functions
Algebraic Expressions
Trigonometric Functions
Exponential Functions

Its acronym, for easier memorization, is LIATE. And so, when you see logarithms, and you are asked to use integration by part, you automatically set your logarithm to u. If there are no logarithms in the integral, then you set your inverse trigonometric function as u. If it's not there you use algebraic expression as u, and so on.

For example, we are asked to integrate \(\int x\ln x dx\). In this example, we have an algebraic expression (which is x) and a logarithmic function (which is ln x). From LIATE, we know that we should set the logarithm as u.

So we let u = lnx dv = xdx
du = \(\large \frac{1}{x} dx\) v = \(\large \frac{x^2}{2}\)

Now, we solve for uv - \(\int vdu\)

\(\large \frac{x^2 \ln x}{2} - \int \frac{x^{\cancel{2}}}{2\cancel{x}} dx\)

\(\large \frac{x^2 \ln x}{2} - \frac{1}{2} \int xdx\)

\(\large \frac{x^2 \ln x}{2} - \frac{1}{2} \frac{x^2}{2}\)


\(\large \frac{x^2 \ln x}{2} - \frac{x^2}{4}\)

Answer 2

what level of calculus is this from? I have heard of it before and when I get in a bind sometimes I hear others mention it. I am taking a final for calculus one tomorrow and we have not covered this topic yet. Is it in calculus two? three? differential equations?

Answer 3

i do not know..it's part of techniques of integration

Answer 4

Calc II

Answer 5

I have a final on those on tuesday =(

Answer 6

did this help? :D

Answer 7

2 be honest.. i didnt read lol- but ill read it since u asked ( o_o )

Answer 8

yay! it might help a lot :)

Answer 9

good luck on your final. I hope calc two isn't much harder than calc one

Answer 10

I guess it helped a little : And Ty :P Im worrying about improper integral and estimation sums: Ty for the post Igbasallote

Answer 11

yay iHelped hehe =)))) i know a little about improper integral...but what's estimation sums?

Answer 12

Finite Riemman sums, I guess. Anyway, improper integrals are interesting. I saw a really cute one not too long ago.. I will see if I can find it.

Answer 13

It like ask u to find how many terms u have to go for a series under certain bouns, like Error is less then .002

Answer 14

Nope, guessed wrong. :-)

Answer 15

AND the professor did 1 problem one it. im little mad lol

Answer 16

oh..i dont think i remember that....lol bmp "cute" this isnt in terms of ffm's version of "cute" is it?

Answer 17

bmp, riemman sum 2, those r trap rules/simps rules. easier

Answer 18

Help me out on some Improper integral? like hints, cuz professor said lots of ppl treated as a normal integral and got the wrong answer..

Answer 19

hmm well i dont have much mastery in improper integral so i cant help you out sorry..but i do know someone who can..she's not yet on though

Answer 20

It's just mind blowing, albeit not too hard. Maybe you have seen it lgbasallote, it's something like integral from 0 to infinity and there were to discontinuities, at 0 and at infinity. The answer was pi.

Answer 21

nope havent seen it..i fear those problems

Answer 22

There is this one: \[ \int_0^{\infty} \frac{2}{x^2 + 1}dx \]for instance. That is not hard, but cute.

Answer 23

But the one I can't remember is cuter.

Answer 24

cal II , then it makes sense that i dont understand it at all haha

Answer 25

@lgbasallote: you forgot something important. the constant + C :P

Answer 26

oh man :/ i always forget that >.< thanks for pointing it out :)

Answer 27

@bmp, the answer for that improper integral is pi....
\[\int\limits_{0}^{\infty}2/x^2+1dx=2\int\limits_{0}^{\infty}1/x^2+1dx=2(\tan^{-1}x |_{0}^{\infty} )=2(\lim_{t \rightarrow \infty}\tan^{-1}t)\]
=2(pi/2)=pi

Answer 28

Yup, but it's quite cool :-). I am mad that I can't remember the cuter one.

Answer 29

@lgba i sthe tutorial ovr?

Answer 30

Thanks for linking me to this @lgbasallote!

Answer 31

hehe you're welcome @Jgip ^_^ you're free to see my other tutorials if you want...idk if some can help you but it's your choice lol...just click some links here..almost all of them are mine..two of them are integral tutorials (one is this and the other is integration by wallis' formula)

http://www.google.com/search?client=safari&rls=en&q=how+to+know+what+to+substitute+as+u+site:openstudy.com&ie=UTF-8&oe=UTF-8#q=note:+this+is+not+a+question+this+is+a+tutorial+site:openstudy.com&hl=fil&client=safari&rls=en&prmd=imvns&ei=WCnTT7WaJuKsiAeZ7dGkAw&start=0&sa=N&bav=on.2,or.r_gc.r_pw.r_qf.,cf.osb&fp=4d551040ed49a312&biw=800&bih=463

Answer 32

cheers I'll keep it in mind :)