Integration by parts is done when you have two terms (with x) being multipled together such as xe^x, xlnx, 3xsinx, 5xcosx, 2x^7sinx and so fourth. You call one of your term 'u' and the other term 'dv'. The term that you pick to be your 'u' should be the term that has a derivative that becomes more simple. In this case with xe^2x, when you take the derivative of x, that becomes more simple. Taking the derivative of e^2x doesnt make it any more simple but since we know how to take its antiderivatve, it's okay to use e^2x as our dv. Now that we have our u and our dv, we take the derivatve of u to find the du and the antiderivative of dv to find v. Then all we have to do is plug in the formula. Note: For a problem such as x^2e^2x you would work it out the same but you would have to do two intergration by parts. I am saying this to let you now that it is possible to have to do multiple integration by parts in a single problem which is why some people prefer tabular integration. I'll show you a problem with two integration by parts in a second but I won't explain it in depth since you can see the explanation for the previous problem. Then I'll show tabluar integration.
Two integration by parts: When you have to do multiple integration by parts, it's usually better to do tabular integration which I'll show in a second if I remember. My teacher never showed or allowed us to use tabular integration on tests.
Oh my genius! I will be back tomorrow. I have to sleep for real now!! But at the mean time what is anti derivative I'm pretty sure I've not hear this term LOL
Tabular integration with the same problem Pick one term as your derivative column and the other term as your antiderivative column. You have to pick a term that's derivative goes to zero for the derivative column or else you can't use tabular integration. Once you have both columns set, keep taking the derivative until you get zero in the derivative column. Then in the antiderivative column, keep taking the antiderivative until you have just as many antiderivatives as derivatives so that the columns are equal in length. Then you are going to flip the sign on every even term and keep the signs on the odd terms the same. Keep the sign the same for the first term, then flip the sign for the second term, keep the sign the same for the third term, flip the sign on the fourth term etc. Then you draw arrows from one term in the derivative to one term lower in the antiderivative column. Then you multiply those all together and add them. So when you see an arrow connecting one term to another, that means to multiply those terms together. What you see in this problem is three arrows so there will be three terms multipled together and then you add all three of those terms. Tabular integration is almost always faster than integration by parts, especially in problems that would require multiple integration by parts. Note that this problem that I did with tabular integration is the exact same problem that just took me two integration by parts and thus was much easier with tabular integration.
Here is a photo of my desk while studying for my math final If anyone wonders where I get all of my paper that I work out problems on that I post to the math thread, it's from the backside of papers such as quizzes, tests, homework, and others which you can see on my desk I will note that I have much more paper than this in the storage slots in my desk which is also stuff such as quizzes etc so what you see is about 1/7 of the paper that I currently have lol
Yea my physics teacher assigns 3 homeworks a week and each homework assignment is up to 30 questions and I print them all out online in my dorm which has free unlimited printing. I get more than 5 times more physics problems weekly for homework than calculus problems. And the physics problems are a lot harder also -_- that's where all of my paper comes from, I use the backs of my physics homework since its all online and I get to keep it.