Fundamental Theorem of Calculus

In the previous lesson, we learned how to find the integral of the function $y=2x$. We were also introduced to a new concept - the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus considers the following situation, $$\frac{d f(x)+C}{dx} = F(x)$$ where $C$ represents any constant (remember - the value of $C$ can be anything, because the derivative of a constant is 0.)

It states that we can perform the following algebra on the above: $$d f(x)+C =F(x) dx$$ $$\int ( d f(x)+C) = \int F(x) dx$$ $$f(x)+C = \int F(x) dx$$

This is because differentiation ($d$) and integration ($\int dx$) are inverse functions.

We can say that again - differentiation and integration are inverse functions.

This is an incredibly useful piece of information -  it allows us to compute the integral of so many functions! For example:

$$\frac{d \sin x}{dx} = \cos x,$$which tells us that

$$\int \cos x\ dx = \sin x.$$ Similarly, since $$\frac{d \cos x}{dx} = - \sin x,$$ $$\int \sin x\ dx = -\cos x.$$Notice how we moved the negative sign from the RHS to the LHS - this is because $-1$ is a constant, and constants can freely travel in or out of an integral.

What we just did is also called taking the antiderivative.

Take a trip back to derivatives and look for all of the functions we differentiated. How can you use this information to integrate our answers?

Can you make use of the methods of differentiation to solve even more integrals?