# Integration by Parts

#### alane1994

##### Active member
When integrating by parts the formula is

$$\int{u} dv=uv-\int{v} du$$

I understand where to get the u, and du... but where does one get the v?

#### MarkFL

Staff member
You get the v from:

$\displaystyle \int\,dv$

#### Chris L T521

##### Well-known member
Staff member
When integrating by parts the formula is

$$\int{u} dv=uv-\int{v} du$$

I understand where to get the u, and du... but where does one get the v?
For example, consider the integral

$\int e^x\sin x\,dx$

The general rule for finding your $u$ is given by the anagram LIPET, where it goes from highest priority to lowest priority.

L: Logarithm
I: Inverse Trigonometric
P: Polynomials/Algebra
E: Exponential
T: Trigonometric

In the case that you only have exponential and trig, either one can be your $u$.

So, for example, if your integrand contains inverse trig and exponential functions, you let $u$ be the inverse trig function.

In the particular example I gave, we only have exponential and trigonometric functions in the integrand, so we can pick either term to be our $u$. Without loss of generality, say $u=e^x$. Now, $\,dv$ is everything else; that is, $\,dv=\sin x\,dx$. Finding $\,du$ is straightforward (we just differentiate to get $\,du=e^x\,dx$). To find $v$, you need to integrate the $\,dv$ term, that is, $v=\int\,dv=\int \sin x\,dx=-\cos x$.

So, to find $v$ in general from integration by parts, you always end up integrating the value you have for $\,dv$.

I hope this makes sense!

EDIT: Ninja'd

Staff member

Staff member

#### MarkFL

Staff member
This leads me to believe that perhaps the order of the last two may depend on the integrand being considered. I am glad to know there is a variation, and that the rule I was taught has some exceptions.

#### Jameson

Staff member
I was never taught that but wish I had. After doing a bunch of these kinds of integrals you just develop an intuition about what to choose for $u$ and $dv$ but before that point it can be really tricky to do, so I think the acronym you guys mentioned is a great tool for making this topic quicker to master.

#### Chris L T521

##### Well-known member
Staff member
I was never taught that but wish I had. After doing a bunch of these kinds of integrals you just develop an intuition about what to choose for $u$ and $dv$ but before that point it can be really tricky to do, so I think the acronym you guys mentioned is a great tool for making this topic quicker to master.
I was never taught this when I took calculus in high school; I only found out about this anagram when I worked as a tutor at a community college. When I saw it, I was like...woah...this simplifies the process quite a lot. Since then, I've told my students about it and it has made doing IBP problems easier for them.

#### Opalg

##### MHB Oldtimer
Staff member
I was never taught an acronym for this. I was given a general rule of thumb that went like this. Many functions get simpler when you differentiate them, most functions get more complicated when you integrate them. When integrating by parts, you want to differentiate the part that looks most likely to become simpler when differentiated, and you should integrate the part that looks least likely to get more complicated when integrated.

#### topsquark

##### Well-known member
MHB Math Helper
I never learned it with the table, so I'm grateful for Chris to mention it.

This didn't bother me so much...after all there are only so many integration problems that you have to deal with. But when he started doing it with differential equations I got a bit confused!

-Dan

#### Chipset3600

##### Member
I'm learning Integration by Parts now in Calculus 1, and at first I was very confused in choosing [TEX]u[/TEX] and the [TEX]dv[/TEX], i learned the "L I A T " in youtube and has helped me a lot

#### alane1994

##### Active member
For example, consider the integral

$\int e^x\sin x\,dx$

The general rule for finding your $u$ is given by the anagram LIPET, where it goes from highest priority to lowest priority.

L: Logarithm
I: Inverse Trigonometric
P: Polynomials/Algebra
E: Exponential
T: Trigonometric

In the case that you only have exponential and trig, either one can be your $u$.

So, for example, if your integrand contains inverse trig and exponential functions, you let $u$ be the inverse trig function.

In the particular example I gave, we only have exponential and trigonometric functions in the integrand, so we can pick either term to be our $u$. Without loss of generality, say $u=e^x$. Now, $\,dv$ is everything else; that is, $\,dv=\sin x\,dx$. Finding $\,du$ is straightforward (we just differentiate to get $\,du=e^x\,dx$). To find $v$, you need to integrate the $\,dv$ term, that is, $v=\int\,dv=\int \sin x\,dx=-\cos x$.

So, to find $v$ in general from integration by parts, you always end up integrating the value you have for $\,dv$.

I hope this makes sense!

EDIT: Ninja'd
That helps alot! Thank you very much! The terminology and way that my professor explained it did not help very much at all. I appreciate all the help.

#### skatenerd

##### Active member
Just a innocent passer-by here. I also wanted to thank you both for the acronyms...I too was just taught in Calculus II to basically develop an intuition for IBP, and in cases when my intuition fails me to just try different options and see which work and which don't. I'm excited to try LIPET and LIATE.