#Khan academy ap calculus bc how to

U = g(x), if the range is some interval and f(x) is continuous over that interval then, I'll give an alternative formulation of the rule. u substitution is another method of evaluating an integral in an attempt to transform an integral that doesn't match a known integral rule into one that does. There are a bunch of different methods and approaches to solve integrals, sometimes multiple methods can be used, sometimes only one can solve it exactly, and sometimes exact equations don't exist leading to numerical approximations. Anti-derivatives require many more rules and methods to evaluate if it can be evaluated in terms of an exact function. While derivatives tend to be evaluated exactly, using a far more limited set of rules. If you were referring to something different than what I just showed, please tell me by commenting on this answer.Īnti-derivatives are harder than derivatives. You can't just get rid of the dx because it needs to cancel out the the dx in the derivative of the function. I think that's Martin Gardner and Sylvanus B. The integral sums up all the differentials of a variable and makes it that variable. If so, do you agree that the (dx)s cancel out it the integral? If so, do you agree that y' equals dy/dx? I guess why not the letter 'u' :)'ĭo you agree that this is what an indefinite integral does? Why the letter 'u'? Well, it could have been anything, but this is the convention. Over time, you'll be able to do these in your head without necessarily even explicitly substituting.

U-substitution is very useful for any integral where an expression is of the form g(f(x))f'(x)(and a few other cases).

It is essentially the reverise chain rule. 'U-substitution is a must-have tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). However, even with integration by parts taken together the two substitutions must completely replace the original variable, with no leftovers allowed. In that case, you do two u-substitutions (but you call them by different variables). It isn't really an exception, but you can sort of have leftover bits when you do integration by parts. ∫ cos (x²) dx is nightmarishly difficult (getting into something called Fresnel integrals). ∫ (x)∙cos(x²) dx is very easy to integrate but the very similar looking With integration, being close to a standard form is not good enough: you must have an exact match. So, the answer is, no, you cannot do u-substitution that way. And remember du is the derivative of whatever you called u, it is NOT just some notation.

#Khan academy ap calculus bc how to

Where f(u) du is something you know how to integrate. The purpose of u substitution is to wind up with ∫ f(u) du A valid substitution, generally speaking, requires that ALL references to the original variable be replaced ESPECIALLY including its dx (or whatever the variable is). Sorry, that is ordinarily not a valid means of substitution.