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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>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

∫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? • 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)? • 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.) • 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
``         xlog(x) = ∫ 1/t dt,    x > 0.         1``