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### Course: Class 12 math (India)>Unit 9

Lesson 2: Indefinite integrals intro

# Reverse power rule

Can you find a function whose derivative is x^n? Created by Sal Khan.

## Want to join the conversation?

• Why didn't he use quotient rule here?
• The reason is that n should NOT be viewed as a variable. even though we do not know what n is, we treat n as a constant. He is simply deriving the formula for integrating x raised to some exponent. The idea is that the formula works for any exponent, but when you actually do a specific problem n would be an actual number. So the quotient rule is not used because x is actually being divided by a constant, we just do not know what the constant is because the idea is that it could be any number and the formula would still work.
• Here's a question, at , is it truly necessary for you to factor out the five before continuing the problem? Or can you simply take the anti derivative of the expression with the 5 in the expression?
• Since the 5 is a constant, you can factor it out or leave it inside the integral. It won't change the answer in the end. However, it is common practice to pull constants out of the integral because it (usually) makes taking the integral easier to read and you can find any mistakes easily if you made them.

tl;dr version, it's completely optionally
• What is the significance of the dx at the end of ∫(f(x))dx. I understand that dx means an infinitesimally small change in x, but what role does it play (or maybe I should ask, what does it represent) in the integration? Thanks.
• To add to what redthumb.liberty said, which is quite true, there is something that beginning integral calculus students often get wrong regarding the dx.

The `dx` is not just some notation that you can tack on. You may interpret the `dx` as the derivative of the variable `x`. This gets important when you start learning standard integral forms.

For example, here is a standard integral form:
`∫ cos (u) du = sin (u) + C`
So, some students will incorrectly see:
`∫ cos (x²) dx` and say its integral must be `sin (x²) + C`. But this is wrong. Since you are treating x² as the u, you must have the derivative of x² as your du. So, you would need `2xdx = du`. Thus, it is
`∫ (2x)cos (x²) dx = sin (x²) + C`
Whereas, `∫ cos (x²) dx` because it doesn't have the derivative of `x²`, is an advanced level integral that couldn't possibly be solved by any method taught in an introductory integral calculus course.
• Is there a name for the symbol used for representing the anti derivative?
• Well, there are two ways of representing the antiderivative.
Generally, the lowercase letter is used for functions; e.g.: f(x). The letters here should also be italicized, but Khan Academy does weird things with the formatting if I were to try to do that.
Anyways, the antiderivative of f(x) is often written as F(x). Thus, F'(x) = f(x). This really cannot be used for anything other than indeﬁnite integrals (which is what antiderivatives are).

The integral sign, ∫ has a bit more of a story for it. This symbol is more speciﬁc to integrals. When this symbol is used for an indeﬁnite integral, one could almost say that an antiderivative is being written in the form of an integral.
A deﬁnite integral basically ﬁnds the sum of an inﬁnite number of parts (learn about that in the next section). Because of this, it would make sense to use an S as the symbol for integration--- S for sum, just like Σ (a Greek S) is used for summation.
Back in the day (over a century ago), English (and other languages) used two forms of a lowercase S. There was a long S, written as ſ (basically an f without the cross-stroke), and there was a terminal S, written as s. The long S was used at the beginning and the middle of a word. To illuſtrate this, I will ſtart uſing it. The terminal S was uſed at the ends of words. Now, perhaps you’ll notice that an italicized f looks kind of like an integral ſign with a croſs-ſtroke. Well, when you italicize an ſ, you baſically get an integral ſign. This is actually where it comes from. In fact, the integral ſign is really the only place where the long S is even uſed anymore.

The integral sign is an italicized long S---the S stands for sum.

Juſt a kind of ſide-bar, if you are familiar with German, you’ve probably ſeen that they have an S symbol that looks kind of like a B. Well, look at the long S and terminal S next to each other: ſs. Now, imagine them being merged into one character: ß. There you go!
• At : why don't you differentiate n+1?
• Because n isn't changing with respect to x.
• If n cannot be -1, how will you integrate x^(-1), aka 1/x?
• The actual antiderivative of 1/x is the absolute value of ln|x|, commonly mistaken with ln(x). This is due to the end behavior of the graph of the natural log of x and the domain and range of the graph. According to the AP course, usually ln|x| is accepted. I hope this helps, and please do feel free to ask further questions as needed.
• This has no relation to the video,but it was on my mind What is the inverse of
x^x(x powered x)
Plz show some steps to proceed
• That is an advanced-level problem that requires the Lambert W-function (also known as the product log) to solve

Let us define a function of x, W(x) such that
W(xe^(x)) = x
(This is known as the Lambert-W function or product log and there are a variety of equivalent ways it can be defined.)

y = x^x
Invert: x = y^y
log x = y log y
note that y = e^(log y)
log x = log y (e^(log y))
W(log x) = W{log y (e^(log y))}
W(log x) = log y
e^W(log x) = y
Note: This can be rewritten as y = log (x) / W(log x)
Writing it this way requires using a common property of the W function:
e^[W(u)] = u / W(u)
• What happens if you have an original function f(x) = x^1. The derivative of this would be just x^0, which is the same as f'(x) = 1. So how do you find the integral of f'(x)? Do you just say f'(x) * dx, and then put (x^1)/1 in the function?
• Yep, just remember that if it is an indefinite integral, you would also have the + C at the end of it.
``f(x) = xf'(x) = 1 dxintegral of f'(x) dx = x + C``
• Do we always need to write down the constant 'c', right after the equation? If this is so, then why can't we write down the answer as "-5x^-1" instead of "-5x^-1 + c"?
• So what is the antiderivative of x^n, when n is -1?
(1 vote)
• ln x + c.
The derivative of ln x is x^-1, which you can find through implicit differentiation.