It might be a silly question, but, I don't have to put a constant to each operation? I saw that he correct himself in the video, but, he put just one constant for two operations, or that C is something like A + B ( the A for the sinx anti-derivative and the B for the cosx anti-derivative) ? Thanks in advance

Short answer: yes. You could add a constant on each term, but since they're arbitrary, they can all be added together into a single, arbitrary constant.

You said that we learned about the natural log in the last video, which video are you talking about?

It's actually in the next video, the Antiderivative of x^-1.

at 3:33 you said always remember about constant "c",its important . why it is?

Consider `∫ 1/2x dx`, without using the constant integration. *Method 1* `∫ 1/2x dx` `= 0.5 ∫ 1/x dx` `= 0.5 ln(x)` ...(1) *Method 2* Let `u = 2x => du/dx = 2 => dx = du/2` `∫ 1/2x dx` `= ∫ 1/2u du` `= 0.5 ∫ 1/u du` `= 0.5 ln(u)` `= 0.5 ln(2x)` ...(2) From (1) we have `∫ 1/2x dx = 0.5 ln(x)`. From (2) we have `∫ 1/2x dx = 0.5 ln(2x)`. *Conclusion*: `x = 2x`? Not quite. Where did we go wrong? The `+C`. The results can be explained by using a property of logarithms- `ln(ab) = ln(a) + ln(b).` So `0.5 ln(2x) = 0.5[ln(2) + ln(x)] = 0.5 ln(x) + 0.5 ln(2) = 0.5 ln(x) + C`. This is why the `+C` is very important.

what is difference between indefinite integral and definite integral?

the indefinate integral isn't between two set values and the definate integral is

why is it the natural log of the absolute value of a and not ln(a)?

Due to the fact that the result of taking the ln of a negative value is undefined. The absolute value application allows negative values to be defined as well since it turns negative values into positive. In conclusion the absolute value application gives the anti-derivative the same domain as 1/x.

What is e? It always pops up in the mathematical world, but I can never figure out what it is. Has Khan made any videos on it?

It actually pops up all over the natural world. It's a somewhat mysterious constant called Euler's number (~ 2.718). It's also the implicit base for natural logarithm (Ln) and useful due to its properties. Yes, Khan made videos on it, look for compound interest and e.

isn't the derivative of sin(t) = -cos(t)?

The anti-derivative of sin(t) is -cos(t) but the derivative of sin(t) is cos(t)

It's pretty easy to see that the derivative of -cosx is sinx, but how can you prove that the only anti derivative of sinx is -cosx?

It is not; adding any constant to `_-cos_` furnishes yet another antiderivative of `_sin_`. There are in fact infinitely many functions whose derivative is `_sin_`. To prove that two antiderivatives of a function may only differ by a constant, follow this outline: suppose a function `ƒ` has antiderivatives `F` and `G`. Define a function `H` by `H = F - G`. Conclude that `H' = 0`, so that `H` is a constant; `F - G = C` holds for some constant `C`. Thus `F = G + C`. It is not hard to make this "proof" rigorous, and I suggest you do so. (Note: when we conclude from the fact that `H'` is zero that `H` is constant, we actually use the mean value theorem.)

what is the natural log? in which video ?? I am so confused . I have so many missing stuff!

There are multiple ways to define the natural logarithm. One way is to define a function `log: (0, +∞) -> R`, called the _natural logarithm_, by the integral ``` x log(x) = ∫ 1/t dt, x > 0. 1 ``` For a treatment of this function using this definition, see, e.g., _Calculus, Volume I_ (2nd ed., Apostol) or _Introduction to Analysis_ (Rosenlicht).

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AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 6

Lesson 9: Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals

Indefinite integrals of sin(x), cos(x), and eˣ

Name: Indefinite integrals of sin(x), cos(x), and eˣ
Uploaded: 2013-01-28T16:59:05Z
Description: ∫sin(x)dx=-cos(x)+C, ∫cos(x)dx=sin(x)+C, and ∫eˣdx=eˣ+C. Learn why this is so and see worked examples.

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∫sin(x)dx=-cos(x)+C, ∫cos(x)dx=sin(x)+C, and ∫eˣdx=eˣ+C. Learn why this is so and see worked examples. Created by Sal Khan.

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Nathan Fernandes
Posted 11 years ago. Direct link to Nathan Fernandes's post “It might be a silly quest...”
It might be a silly question, but, I don't have to put a constant to each operation?

I saw that he correct himself in the video, but, he put just one constant for two operations, or that C is something like A + B ( the A for the sinx anti-derivative and the B for the cosx anti-derivative) ?

Thanks in advance
Button navigates to signup pageComment on Nathan Fernandes's post “It might be a silly quest...”
(77 votes)
Answer
- Michael Hough
  Posted 11 years ago. Direct link to Michael Hough's post “Short answer: yes. You c...”
  Short answer: yes.
  
  You could add a constant on each term, but since they're arbitrary, they can all be added together into a single, arbitrary constant.
  Button navigates to signup page
  (173 votes)
briana collins
Posted 11 years ago. Direct link to briana collins's post “You said that we learned ...”
You said that we learned about the natural log in the last video, which video are you talking about?
Button navigates to signup pageButton navigates to signup page
(53 votes)
Answer
- Tyler
  Posted 11 years ago. Direct link to Tyler's post “It's actually in the next...”
  It's actually in the next video, the Antiderivative of x^-1.
  Button navigates to signup page
  (67 votes)
shariar
Posted 11 years ago. Direct link to shariar's post “at 3:33 you said always ...”
at
3:33
you said always remember about constant "c",its important . why it is?
Button navigates to signup pageButton navigates to signup page
(15 votes)
Answer
- ↑←xylon97→↓
  Posted 11 years ago. Direct link to ↑←xylon97→↓'s post “Consider `∫ 1/2x dx`, wit...”
  Consider ∫ 1/2x dx, without using the constant integration.
  
  Method 1
  ∫ 1/2x dx
  = 0.5 ∫ 1/x dx
  = 0.5 ln(x) ...(1)
  
  Method 2
  Let u = 2x => du/dx = 2 => dx = du/2
  ∫ 1/2x dx
  = ∫ 1/2u du
  = 0.5 ∫ 1/u du
  = 0.5 ln(u)
  = 0.5 ln(2x) ...(2)
  
  From (1) we have ∫ 1/2x dx = 0.5 ln(x).
  From (2) we have ∫ 1/2x dx = 0.5 ln(2x).
  
  Conclusion: x = 2x?
  Not quite.
  
  Where did we go wrong? The +C.
  The results can be explained by using a property of logarithms-
  ln(ab) = ln(a) + ln(b).
  So 0.5 ln(2x) = 0.5[ln(2) + ln(x)] = 0.5 ln(x) + 0.5 ln(2) = 0.5 ln(x) + C.
  This is why the +C is very important.
  Comment on ↑←xylon97→↓'s post “Consider `∫ 1/2x dx`, wit...”
  (65 votes)
Min ChanHong
Posted 11 years ago. Direct link to Min ChanHong's post “what is difference betwee...”
what is difference between indefinite integral and definite integral?
Button navigates to signup pageButton navigates to signup page
(15 votes)
Answer
- Lisa Satpathy
  Posted 11 years ago. Direct link to Lisa Satpathy's post “the indefinate integral i...”
  the indefinate integral isn't between two set values and the definate integral is
  Button navigates to signup page
  (27 votes)
Elisha Day
Posted 11 years ago. Direct link to Elisha Day's post “why is it the natural log...”
why is it the natural log of the absolute value of a and not ln(a)?
Button navigates to signup pageButton navigates to signup page
(14 votes)
Answer
- Ronie Islam
  Posted 11 years ago. Direct link to Ronie Islam's post “Due to the fact that the ...”
  Due to the fact that the result of taking the ln of a negative value is undefined. The absolute value application allows negative values to be defined as well since it turns negative values into positive. In conclusion the absolute value application gives the anti-derivative the same domain as 1/x.
  Comment on Ronie Islam's post “Due to the fact that the ...”
  (11 votes)
Rohan Khajuria
Posted 11 years ago. Direct link to Rohan Khajuria's post “What is e? It always pops...”
What is e? It always pops up in the mathematical world, but I can never figure out what it is. Has Khan made any videos on it?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- bruno.garcia
  Posted 11 years ago. Direct link to bruno.garcia's post “It actually pops up all o...”
  It actually pops up all over the natural world. It's a somewhat mysterious constant called Euler's number (~ 2.718). It's also the implicit base for natural logarithm (Ln) and useful due to its properties. Yes, Khan made videos on it, look for compound interest and e.
  Button navigates to signup page
  (12 votes)
tom gibbs
Posted 11 years ago. Direct link to tom gibbs's post “isn't the derivative of s...”
isn't the derivative of sin(t) = -cos(t)?
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(3 votes)
Answer
- manil_trivedi
  Posted 10 years ago. Direct link to manil_trivedi's post “The anti-derivative of si...”
  The anti-derivative of sin(t) is -cos(t) but the derivative of sin(t) is cos(t)
  Comment on manil_trivedi's post “The anti-derivative of si...”
  (9 votes)
Pierre-Olivier
Posted 10 years ago. Direct link to Pierre-Olivier's post “What is the antiderivativ...”
What is the antiderivative of e^(4x)? None of sal's examples for antiderivatives include chain rule stuff. I read everywhere that the antiderivative of e^(4x) is e^(4x)/4. It makes sense, because if you asked me to find the derivative of e^(4x)/4, I would do the chain rule by multiplying that by 4 (which is the derivative of 4x), which would give me 4e^(4x)/4, equaling the original e^(4x). But I don't understand how to get back there with the antiderivative.
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(4 votes)
Answer
Raviv Sarch
Posted 9 years ago. Direct link to Raviv Sarch's post “It's pretty easy to see t...”
It's pretty easy to see that the derivative of -cosx is sinx, but how can you prove that the only anti derivative of sinx is -cosx?
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(2 votes)
Answer
- Qeeko
  Posted 9 years ago. Direct link to Qeeko's post “It is not; adding any con...”
  It is not; adding any constant to -cos furnishes yet another antiderivative of sin. There are in fact infinitely many functions whose derivative is sin.
  
  To prove that two antiderivatives of a function may only differ by a constant, follow this outline: suppose a function ƒ has antiderivatives F and G. Define a function H by H = F - G. Conclude that H' = 0, so that H is a constant; F - G = C holds for some constant C. Thus F = G + C. It is not hard to make this "proof" rigorous, and I suggest you do so.
  
  (Note: when we conclude from the fact that H' is zero that H is constant, we actually use the mean value theorem.)
  Comment on Qeeko's post “It is not; adding any con...”
  (8 votes)
mira_ocean_sky
Posted 9 years ago. Direct link to mira_ocean_sky's post “what is the natural log? ...”
what is the natural log? in which video ?? I am so confused . I have so many missing stuff!
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Qeeko
  Posted 9 years ago. Direct link to Qeeko's post “There are multiple ways t...”
  There are multiple ways to define the natural logarithm. One way is to define a function log: (0, +∞) -> R, called the natural logarithm, by the integral
  x log(x) = ∫ 1/t dt, x > 0. 1
  
  For a treatment of this function using this definition, see, e.g., Calculus, Volume I (2nd ed., Apostol) or Introduction to Analysis (Rosenlicht).
  Comment on Qeeko's post “There are multiple ways t...”
  (2 votes)

Video transcript

I thought I would do a few more examples of taking antiderivatives, just so we feel comfortable taking antiderivatives of all of the basic functions that we know how to take the derivatives of. And on top of that, I just want to make it clear that it doesn't always have to be functions of x. Here we have a function of t, and we're taking the antiderivative with respect to t. And so you would not write a dx here. That is not the notation. You'll see why when we focus on definite integrals. So what's the antiderivative of this business right over here? Well, it's going to be the same thing as the antiderivative of sine of t, or the indefinite integral of sine of t, plus the indefinite integral, or the antiderivative, of cosine of t. So let's think about what these antiderivatives are. And we already know a little bit about taking the derivatives of trig functions. We know that the derivative with respect to t of cosine of t is equal to negative sine of t. So if we want a sine of t here, we would just have to take the derivative of negative cosine t. If we take the derivative of negative cosine t, then we get positive sine of t. The derivative with respect to t of cosine t is negative sine of t. We have the negative out front. It becomes positive sine of t. So the antiderivative of sine of t is negative cosine of t. So this is going to be equal to negative cosine of t. And then what's the antiderivative of cosine of t? Well, we already know that the derivative with respect to t of sine of t is equal to cosine of t. So cosine of t's antiderivative is just sine of t-- so plus sine of t. And we're done. We've found the antiderivative. Now let's tackle this. Now we don't have a t. We're taking the indefinite integral with respect to-- actually, this is a mistake. This should be with respect to a. Let me clean this up. This should be a da. If we were taking this with respect to t, then we would treat all of these things as just constants. But I don't want to confuse you right now. Let me make it clear. This is going to be da. That's what we are integrating or taking the antiderivative with respect to. So what is this going to be equal to? Well once again, we can rewrite it as the sum of integrals. This is the indefinite integral of e to the a da, so this one right over here-- a d I'll do it in green-- plus the indefinite integral, or the antiderivative, of 1/a da. Now, what is the antiderivative of e to the a? Well, we already know a little bit about exponentials. The derivative with respect to x of e to the x is equal to e to the x. That's one of the reasons why e in the exponential function in general is so amazing. And if we just replaced a with x or x with a, you get the derivative with respect to a of e to the a is equal to e to the a. So the antiderivative here, the derivative of e to the a, the antiderivative is going to be e to the a. And maybe you can shift it by some type of a constant. Oh, and let me not forget, I have to put my constant right over here. I could have a constant factor. So let me-- always important. Remember the constant. So you have a constant factor right over here. Never forget that. I almost did. So once again, over here, what's the antiderivative of e to the a? It is e to the a. What's the antiderivative of 1/a? Well, we've seen that in the last video. It is going to be the natural log of the absolute value of a. And then we want to have the most general antiderivative, so there could be a constant factor out here as well. And we are done. We found the antiderivative of both of these expressions.