You're running late for work.
Your smart watch tells you the speed at which you're biking and the absolute distance from your home.
But in reality, you've covered a greater distance, since the road from your home to your workplace looks like the letter S. (Also, you've had to veer for innumerable people on electric kick bikes.)
How can you figure out the actual distance?
In this case, the derivative corresponds to your velocity. Since the velocity is the derivative of the position, then the position must be what you get by reversing the differentiation.
The derivative is for the antiderivative what yin is for yang
In general terms; whatever you get by reversing the differentiation is called an antiderivative.
Antiderivatives are usually denoted by capital letters. If the velocity is given by , the position becomes , and:
regardless of the value of , we can construct an infinite number of antiderivatives by tweaking .
The general form of the antiderivative, , is so important that it has an own name: the indefinite integral. It is denoted by the symbol, meaning that:
The antiderivative is the evil twin of the derivative. The antiderivative wants to undo everything the derivative does, reversing the effect of the differentiation.
The antiderivative is the evil twin of the derivative
The antiderivative was born first, and he prides himself on being the older twin. He thinks he's the original function, and that the derivative just a boring spin-off. In the process of differentiation, something is lost.
If the derivative is called , the antiderivative, being the older twin, is called . The antiderivative is defined as the function such that:
By differentiating the antiderivative, you get .
Finding the antiderivative is largely a matter of studying examples. Once you've built an intuition, you work out the antiderivative by guesswork.
What if ?
The derivative of is . So is the derivative of . Thus, there are several antiderivatives, like:
In fact, you can construct an infinite number of antiderivatives on the form by tweaking the value of the constant term.
This is why the antiderivative thinks he's got more character. He's more information-rich.
Another example. This time is . Here's one suggestion:
Verify that it's an antiderivative by differentiating . Did you get ?
The indefinite integral can be thought of as an umbrella. It kind of "covers" all possible antiderivatives.
For the indefinite integral, we use this
symbol. Once you've got a hang of it, it's quite satisfying to write out.
The antiderivative of a function is:
So the whole thingy is the antiderivative. And when you differentiate the antiderivative, you end up with again.
The constant is what sets the indefinite integral apart from the antiderivative. In that sense, the indefinite integral is just the generalized antiderivative.
The indefinite integral is the generalized antiderivative
Time for an example! You're out for a car ride. Let be your velocity as a function of the time . If you differentiate the position, you'll get the velocity. Conversely, the indefinite integral of the velocity corresponds to the position. Hence, your position is:
The value of the constant depends on where you started. If is the position in relation to your home, and you started on your driveway, . However, if you started driving from the pub, located some metres from your home, then .