# Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is broken into two parts , which both basically mean the same thing. Firstly, given any differentiable function $$f(x)$$, there is an anti-derivative of $$f'(x)$$ to get back $$f(x)$$. On the other hand, Given the anti-derivative of $$f(x)$$, we can differentiate it to get back $$f(x)$$

For instance, $$f(x)=x^2$$

$$f'(x)=2x$$

$$\int f'(x)dx= \int 2xdx = x^2+C$$

When $$C=0$$, $$\int f'(x)dx=x^2=f(x)$$

On the other hand, $$\int f(x)dx = \frac{1}{3} x^3+C$$

$$(\int f(x) dx)'=(\frac{1}{3} x^3+C)'=x^2=f(x)$$

You are welcome to prove it below.

Note by Aloysius Ng
3 years, 6 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$