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### Course: AP®︎ Calculus BC (2017 edition)>Unit 9

Lesson 4: Indefinite integrals of sin(x), cos(x), eˣ, and 1/x

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

∫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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• 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
• 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)
• 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.
• Life advice : Always remember the constant!!
• huh yeah. that can come back and sting u if u forget it
• 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.)