# High School Calculus/The Fundamental Theorems of Calculus

### The Fundamental Theorems of Calculus[edit]

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

An antiderivative of the function is any function, often denoted by , such that

.

Let's do some practice on this

**Ex.1**

Find the antiderivative of is

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

In general, the antiderivative of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle x^k}**
is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \frac {1}{k} * x^{k +1}}**

**Ex. 2**

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**

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.