Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.
In this lesson, see how to find the integral of cos(x) using well-known derivatives and the fundamental theorem of calculus. We also examine how we can use this theorem to see a connection between derivatives and integrals.
Steps to Solve
We want to see how to find the integral of cos(x). To find this integral, we make use of the first part of the fundamental theorem of calculus. You may be thinking that this theorem sounds a bit daunting. After all, it is the 'fundamental' theorem of calculus! Well, don't let that intimidate you. The good news is that, though this is an essential theorem in calculus, it is actually pretty straight forward.
First, let's take a look at this theorem in technical terms.
Hmm, looking at it like that, it's easy to see why so many people would think this does look intimidating! Let's put this in simpler terms. Basically, if a is the derivative of b, then the integral of a is b + C, where C is a constant.
Alright, now that we understand this theorem, let's use it to find the integral of cos(x). According to the theorem, the integral of cos(x) will be equal to the function that has cos(x) as its derivative plus a constant. Great! All we need to know is what function has cos(x) as its derivative, and luckily, it is well known that the derivative of sin(x) is cos(x). We've got all the information we need!
By the fundamental theorem of calculus and the fact that the derivative of sin(x) is cos(x), we have that the integral of cos(x) is sin(x) + C, where C is a constant.
The Fundamental Theorem of Calculus
As we just saw, it is extremely useful to know the relationships between trigonometric functions and their derivatives. Not only is it important because it makes working with derivatives of these types of functions easier, but it also allows us to find integrals involving these types of functions much more easily, thanks to the fundamental theorem of calculus.
For instance, suppose we wanted to calculate the integral of sec 2 (x). This looks like it would be really complicated to do. However, it just so happens that the derivative of tan(x) is sec 2 (x), so by the fundamental theorem of calculus, we have that the integral of sec 2 (x) is tan(x) + C, where C is a constant. We see that knowing the derivatives of the trigonometric functions makes a seemingly difficult integration problem quite simple.
Based on these things that we've learned, let's take a look at the derivatives of the trigonometric functions.
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That's a lot! As we just saw when finding the integral of cos(x), since the derivative of sin(x) is cos(x), the integral of cos(x) is sin(x) + C. The fundamental theorem of calculus tells us that by knowing the derivatives of these six trigonometric functions, we also know six integrals.
sin(x) + C
cos(x) + C
sec 2 (x)
tan(x) + C
csc(x) + C
sec(x) + C
-csc 2 (x)
cot(x) + C
Again, quite a lot there! Pretty neat and handy, huh? It is an excellent idea to put these derivatives and the fundamental theorem of calculus to memory. After all we've seen, it is easy to see how useful these facts are.
This may leave you wondering if this means that whenever we know a derivative, that fact also gives us an integral. Why, yes! That's exactly what the fundamental theorem of calculus states. Now do you see why it is 'fundamental' to the study of calculus? It's a big one, and because of it, we can now see how intricately connected derivatives and integrals really are.
Let's take a couple of moments to review what we've learned about solving the integral of cos(x). To find this integral, we make use of the first part of the fundamental theorem of calculus. In this theorem, we need to suppose that 'f' is continuous on 'a' and 'b'. From there, we follow two steps to explain the theorem:
Keeping the fundamental theorem in mind, as well as all the trigonometric functions, you should now be able to solve any integral problem, including the integral of cos(x).
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