So, what are indefinite integrals? When you learned derivatives you were supposed to solve the following problem. Given the function f(x), find the function F(x) = f'(x). With indefinite integrals we'll solve the reverse problem.
For example, given the function:
We may ask for a function F(x) whose derivative is f(x). With your vast knowledge of derivatives you may quickly (or not) point out that:
And you would be right. The function F(x) is called a primitive of f(x).
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Now, let's see... What about the functions:
We also have:
So, F1 and F2 are also primitives of f(x). In fact, any function of the form:
Is a primitive of f(x), where C is any constant.
If we want to find all the primitives of a function, we just need to find one primitive. Then, all the other primitives will be the one we found plus a constant.
The indefinite integral of a function is just the set of all the primitives of that function. This is a very simple idea, but we use a very fancy notation.
To denote the indefinite integral of a function f(x) we write:
This is read "the indefinite integral of f(x) with respect to x". The curvy symbol is the symbol for the integral. It was invented by Leibniz. Then we have f(x), the function we're "integrating".
We use "dx" to specify the variable (x in this case).
For example, if "y" was the variable, we would have:
It will become clearer to you why we use these symbols, or even why we call this thing integral when you learn about definite integrals.
For now, we'll learn how to find them.
To find an indefinite integral we just need to find any primitive. In the first example we had:
And we said a primitive of this function is:
The indefinite integral of f(x) is the set of all functions of the form:
And we simply write this as:
We found this using our knowledge of derivatives. And this is generally how we'll do it.
And that's the first key. Every formula for derivatives is also a formula for integrals. You just need to rewrite it.
For example, given the function:
We know that:
This implies that:
This means that we can adapt our table of derivatives for it to become a table of integrals. We'll at least have a basic table.
We can start with this short table...
You can verify all of these formulas simply differentiating the right side of each equation. You'll get the function inside the sign at the left.
You'll find a table similar to this one in your book. Or maybe your teacher gives you one. You'll be using it intensely.
We can say that integrating a function is the opposite operation to deriving a function. This means that given the indefinite integral of a function:
We know this is also a function. And if we derive this with respect to x, what do we get? We get f(x) back:
This is because of the definition of primitive of a function.
And what do we get if we integrate the derivative of a function? That is:
The definition of primitive of a function is "a function whose derivative is the original function". So, this is simply:
And that is what we mean when we say that integration is the opposite operation to differentiation. This is important, because it allows us to say some interesting things about indefinite integrals.
For example, just as derivatives, integrals have the property of sums:
That means that if we have:
We can find that as the sum:
Also, we can take constants out of the integral sign. For example:
There are of course many other properties of integrals. They are all related to properties of derivatives. To continue your learning of integrals, you may go to any of the following pages:
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