Indefinite integral as anti-derivative
Antiderivatives and indefinite integrals
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- We know how to take derivatives of
- functions. If I apply the derivative operator to
- X^2, I get 2X.
- Now if I also apply the derivative operator to
- X^2+1, I also get 2X.
- If I apply the derivative operator to X^2
- X^2+pi, I also get 2X. The derivative of X^2
- is 2X, the derivative with respect to X of pi of a constant
- is just 0. The derivative with respect to X of 1
- is just a constant, is just 0.
- So once again, this is just going to be equal to 2X.
- In general, the derivative with respect to x of X^2
- plus any constant, any constant, is going to be
- equal to 2X. The derivative of X^2 with respect to X is 2X
- Derivative of a constant with respect to X, a constant does not change
- with respect to X, so it's just equal
- to 0. So you have, you take the derivative, you apply the derivative operator
- to any of these expressions, and you get 2X.
- Now let's go the other way around. Let's think about the antiderivative.
- Anti-derivative. And one way of
- thinking about it, it's the, it's the,
- we're doing the opposite of the derivative operator.
- The derivative operator, you get an expression,
- and you find find this derivative.
- Now what we want to do, is, given some expression,
- we want to find
- what it could be the derivative of.
- What it could be the derivative of.
- So if someone were to tell you,
- were to give you 2X.
- If someone were to say, 2X
- So if someone were to ask you: what is
- what is 2X
- the derivative - derivative -
- of? They're essentially asking you for
- the antiderivative!
- And so you could say: well, 2X is
- the derivative of X^2.
- 2X is the derivative of X^2.
- But you could say: 2X is the derivative of X^2 plus 1.
- You could also say that 2X is the derivative of X^2 plus pi.
- I think you get the general idea.
- So if you wanted to write it in the most general sense.
- you would write:
- that 2X is the derivative of X^2
- plus some constant.
- So this is what you would consider
- the antiderivative of 2X.
- Now that's all nice, but this is kind of
- clumsy to have to write the sentence like this
- so lets come up with some kind of notation
- for the antiderivative.
- And the convention here is to use
- kind of a strange-looking notation.
- Is to use a
- a big, elongated, S-looking thing like that
- and the dx around the function
- you're trying to take the antiderivative
- of. So in this case, it would look something like this.
- This is just
- saying: this is equal to the antiderivative of 2x.
- And the antiderivative of 2X, we have already seen,
- is X^2, is X^2 plus C.
- Now, you might be saying:
- Why do we use this type
- of crazy notation?
- It will become more obvious when we
- study the definitive integral, and areas under curves,
- and taking sums of
- rectangles, in order to approximate the area of the curve.
- Here, it really should just be viewed as a notation for
- antiderivatives.
- And, this notation right over here,
- this whole expression, is called:
- the indefinite integral.
- The indefinite...indefinite...
- indefinite integral of 2X
- which is another way of just saying:
- the antiderivative of
- 2X.
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