High School Calculus/The Fundamental Theorems of Calculus

The Fundamental Theorems of Calculus
In order to understand the fundamental theorem of calculus we must first understand what an Antiderivative is.

An antiderivative of the function $$f(x)$$ is any function, often denoted by $$F(x)$$, such that $$F^{\prime} (x)= f(x)$$.

$$\int f(x) = F(x) + C$$

Let's do some practice on this

Ex.1

Find the antiderivative of $$f(x)=x^3$$ is $$F(x) = \frac {1}{4} x^4 + C$$ The C stands for some constant. The reason for this is when you differentiate the stand alone constants become 0

When you differentiate this problem you will end up with $$x^3$$

In general, the antiderivative of $$x^k$$ is $$\frac {1}{k} * x^{k +1}$$

Ex. 2

$$f(x)= x^2 + 3$$ $$\int x^2 + 3 \operatorname {d}x$$

$$F(x) = \int x^2 \operatorname {d}x + \int 3 \operatorname {d}x$$

$$F(x) = \frac {1}{3} x^3 + 3x + C$$

When dealing with functions that have a plus or minus in them you can integrate the separately to help you out and really focus on what is going on. With enough practice you won't need to do this. Remember to keep the appropriate sign between the integrals.

Ex. 3

$$f(x)=85x^7$$

$$F(x)= \int 85x^7 \operatorname {d}x$$

$$F(x) = 85*\int x^7 \operatorname {d}x$$

$$F(x) = 85[\frac {1}{8} x^8]$$

$$F(x) = \frac {85}{8} x^8 + C$$

What was done here was a constant multiplier was pulled out. When you have a common constant multiplier in a function, you can pull it out of the integral to make it easier to evaluate. Just don't forget to multiply it back in when you are done.