Antiderivatives
Next Lesson
To begin our discussing of integral calculus, we'll talk about anti derivatives. capital F of X is the antiderivative of FX on an interval L. If the derivative capital X equals f of x for all x in the interval. So let's just say we have an F of X a simple f of x like. 2x.
Well, one possible capital F of x would be x squared and practice would work for all real numbers. The derivative of x squared is 2X. But notice we could make another choice. We could choose x squared plus five for example. And of course the derivative of 5 would be 0.
So the derivative of x squared + 5 is also 2 x. And in fact, we could add or subtract any number we wanted and still get the same derivative. And so what this means is the most general form of the antiderivative is x squared + C and C is known. As an arbitrary constant.
And anytime we find a general anti-derivative, we're going to have to include an arbitrary constant. Because, of course, when you take it Derivative, any constant would vanish, it would become zero. So we'd never know what constant would be there. And often it actually turns out in various applications there'll be other conditions which determine the arbitrary constant.
So mostly finding The anti-derivative is just a matter of following the derivative rules backwards. So, for example, if f(x) equals x cubed. Well, we'd like to know what the anti-derivative of that would be? Well, we suspect it will be something involving x to the fourth.
We know that if we just take the derivative Have y equals x to the fourth, what we get is four x cubed. And we don't want that four. So we need to multiply by a factor that cancels it. So we multiply by a quarter. Then for example, capital F of x could be one quarter x to the fourth.
And notice that if we took a derivative of this What we would get is 4 over 4 x cubed, or just x cubed by itself. So, that is the antiderivative, of course we'll need a plus c there. If f of x equals sine x, well, we might ask what has a derivative of sine x? We may remember That the derivative of cosine is negative sine x. So we need to toss in a negative sign.
So if we take a derivative of negative cosine, we will get a positive sign. And so they're the anti derivative general integrator, this negative cosine plus C. So this is just an introduction to the idea of anti derivatives, and I'll show an application in the next video.
Read full transcriptWell, one possible capital F of x would be x squared and practice would work for all real numbers. The derivative of x squared is 2X. But notice we could make another choice. We could choose x squared plus five for example. And of course the derivative of 5 would be 0.
So the derivative of x squared + 5 is also 2 x. And in fact, we could add or subtract any number we wanted and still get the same derivative. And so what this means is the most general form of the antiderivative is x squared + C and C is known. As an arbitrary constant.
And anytime we find a general anti-derivative, we're going to have to include an arbitrary constant. Because, of course, when you take it Derivative, any constant would vanish, it would become zero. So we'd never know what constant would be there. And often it actually turns out in various applications there'll be other conditions which determine the arbitrary constant.
So mostly finding The anti-derivative is just a matter of following the derivative rules backwards. So, for example, if f(x) equals x cubed. Well, we'd like to know what the anti-derivative of that would be? Well, we suspect it will be something involving x to the fourth.
We know that if we just take the derivative Have y equals x to the fourth, what we get is four x cubed. And we don't want that four. So we need to multiply by a factor that cancels it. So we multiply by a quarter. Then for example, capital F of x could be one quarter x to the fourth.
And notice that if we took a derivative of this What we would get is 4 over 4 x cubed, or just x cubed by itself. So, that is the antiderivative, of course we'll need a plus c there. If f of x equals sine x, well, we might ask what has a derivative of sine x? We may remember That the derivative of cosine is negative sine x. So we need to toss in a negative sign.
So if we take a derivative of negative cosine, we will get a positive sign. And so they're the anti derivative general integrator, this negative cosine plus C. So this is just an introduction to the idea of anti derivatives, and I'll show an application in the next video.