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## Calculus 2

### Unit 1: Lesson 11

Indefinite integrals of common functions

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

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.C (LO)
,
FUN‑6.C.1 (EK)
,
FUN‑6.C.2 (EK)
∫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.

## Want to join the conversation?

• 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) ?

You could add a constant on each term, but since they're arbitrary, they can all be added together into a single, arbitrary constant.
• It's actually in the next video, the Antiderivative of x^-1.
• at 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
• 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)
• 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.
• 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.
• 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!
(1 vote)
• why is the integration of. Sin(2x -3) = (-1/2)cos (3-2x ) ; here why is it (3-2x ) instead of (2x -3) ?