Integration by Parts
Integration by Parts
A Guide to Integration by Parts:
Given a problem of the form: ∫f(x)g(x)dx=? Choose variables u and v such that u=f(x) dv=g(x)dx Now the expression becomes: ∫udv=? Which definitely looks easier. Anyway, I gotta run. But good luck!

Integration by parts is an integration strategy that is used to evaluate difficult integrals by trying to find simpler integrals derived from the original. It is commonly a source of confusion or irritation for students when they first learn it, due to the fact that there is really no way to accurately predict the proper u/dv separation just by looking at an integral. Integration by parts requires patience, trial and error, and experience.
Randall shows a somewhat complicated math problem and, in an attempt to "help", simplifies it into a more compact integral. This is the first part of performing integration by parts, which involves the guessing. Having gotten it into integration by parts format, he then leaves without describing the actual solution. The general integral ∫ (u dv) is equal to uv - ∫ (v du) , and this is the more tedious part of the math and where problems will arise if you picked the wrong u and dv at the beginning. The narrator makes a point of leaving here, so we can't ask for help or complain if the choice of u and dv was wrong.
The title text points out that if the integral of x can be divided so that u = x and dv = dx and implying v = x, then it leads to the result (1/2)x². This implies the original integral was just ∫x dx, and not needing integration by parts in the first place. Mathematics teachers and extreme math geeks will also cringe at this answer, however, since an indefinite integral requires an integration constant. The correct answer is actually (1/2)x² + C, as Randall hints. The +C symbolizes that an indefinite integral can be shifted by any constant and still gets the same answer on the reverse derivative . Definite integrals specify a range that they're valid on and thus there is no need to add this constant.
A Guide to Integration by Parts:
Given a problem of the form: ∫f(x)g(x)dx=? Choose variables u and v such that u=f(x) dv=g(x)dx Now the expression becomes: ∫udv=? Which definitely looks easier. Anyway, I gotta run. But good luck!
