A lot of people do integration by parts by defining the variables and and flip them around.
There is a more adult way of doing this, which looks nicer and is a lot faster.
Say we have the integral,
The idea is to turn one of the factors into a derivative. For example, we know that .
Thus, we can think of the integral as,
The next step is to take the function inside the differenciation operator and multiply it with the function unaffected with the differenciation and multiply them together. That is the part that you get.
Thus, we get
The next step is to take the derivative of the function which was unaffected by differenciation and multiply it by the function inside the differenciation sign. This is our part.
In this case we get, .
Here is another example,
Look how fast that is.
Here is another example,
My point is that it is a lot easier to keep track of everything doing integration this way. Because you do not need to go out of your way to write and .
The method is cool and I recently translated to post it in my spanish forum. (Not mine, of course.)
Basically it seems your just skipping the writing of "Let u = ...", and doing that part in your head. I do it often too. (I'm lazy! )
EDIT: No not quite skipping it... But I see what you're doing. It is much faster.
Dear Perfect H:
I like your method of integration by parts a lot. I haven't tested it yet, so I am still left with one question: is this a complete replacement for u substitution? If so I am very pleased!
Another thing I'm wondering is, are there ever cases in which we can't do the first step, i.e. in which we can't turn one of the factors into a (solvable) derivative? If so, what then?
There's also tabular integration (tic-tac-toe method) which makes it easier to work with the cases where you need to apply integration by parts more than once.
I think I've been using this formula wrong. Do we ever actually take the derivative of the thing we originally turned into the derivative? This would be clearer if the example hadn't used a base e exponent; I think that is what may be screwing me up.
I tried to take the antiderivative of:
First I decided that I could take the derivative of . Since it was in the denominator first, I had to turned into a and make it part of the numerator. Then I multiplied that by
For the other side of the integral, I took the derivative of , which, lacking a variable, is 1. That left , which integrates to
That leaves me with
The definite integrals I take are all quite preposterous (e.g. 5.02 billion minus negative infinity), implying my antiderivative is wrong. (My end goal is to find the difference of the 70th and 30th percentiles of X; I assume I'll need the antiderivative for this but I haven't quite worked out the endgame.)
I hate using the formula as well. But my way isn't nearly as formal as TPH's. I just always thought of it as: "the integral of one function times the other, minus the of the same function times the derivative of the other" sounds confusing in words, but it helps my weird mind to remember. i just remember to integrate one function, and that appears in both factors. then i put the other function in the first factor, and its derivative in the second factor.
I found the short form of the formula for integration by parts at Wikipedia.
I think I'm finally starting to get it. The short form is:
Okay ... processing. I think this means that where there is a "u dv" you are actually multiplying variable u by variable dv. Also, dv is the derivative of v. (Why they didn't include a multiplier dot between u and dv is unknown. Edit: it is also unknown why the math script is taking the space out from between u and dv in the above.)
I'm still not sure if I have the correct intepretation here, because in every other case of d_ following an integration symbol the formula did not call for multiplication.
If I'm not mistaken, dx never means multiply, it's just this little thing that follows up integrations or antiderivations (presumably to make the multivariable calculus mavens happy). An example of dx not meaning "multiply by the derivative of x" is two lines up on the Wikipedia page (again "if I'm not mistaken").
I'm really jonesing to get this figured out. Got my fingers crossed.