Antiderivatives...relations? Is their any correlation between a function and it's antiderivative? I know the formula for to get said antiderivative is 
$\int F(x)dx=f(x)+C$ 
Where F(x) is the antiderivative of f(x) and 
where C is a arbitrary constant
What is the relation between a function and it's antiderivative, and how can on find the antidervative of a function. Please explain as thoroughly as you can!
 A: You know what derivatives are hopefully. The formula for a derivative is:
$$\dfrac{d}{dx}f(x)=\lim\limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$$
For example, the derivative of $x^2$ is $2x$. This is written like so:
$$\dfrac{d}{dx}x^2=2x$$
There are other notations too, but this is what I like to use.
The antiderivative is basically the reverse operation. For example, the antiderivative of $2x$ is $x^2$. See the relationship? This is written as:
$$\int 2x \ dx = x^2+C$$
Why the $+C$? We must consider a constant because the derivative of a constant is $0$. So we can say that:
$$\dfrac{d}{dx}x^2+5=2x$$
$$\dfrac{d}{dx}x^2-10^{1000}=2x$$
We can also say that:
$$\int 2x \ dx = x^2+5$$
$$\int 2x \ dx = x^2-10^{1000}$$
To make it shorter and more general, we just write $+C$. Now you may be wondering, "What is the $dx$ for?" The $dx$ is there to show people that we are integrating $2x$ in respect to $x$, and not some other variable like $y$ or $z$. Antiderivatives are useful for finding areas under a curve. The definite integral
$$\int_a^b 2x \ dx$$
Is the area under the line $f(x)=2x$ from $x=a$ to $x=b$. The fundamental theorem of calculus explains how to do this. The theorem states that:

If $G$ is an antiderivative of $f$, then:
  $$\int_a^b f(x) \ dx = G(b)-G(a)$$

Let's use that to find out the value of the definite integral
$$\int_0^2 2x \ dx$$
We know that the antiderivative of $2x$ is $x^2+C$.
$$\int_0^2 2x \ dx=\left[(2)^2+C\right] - \left[(0)^2+C\right]$$
$$=4+C-C$$
$$=4$$
Check for yourself that the area under the line $f(x)=2x$ from $x=0$ to $x=2$ is $4$. Amazing, right?
To find an integral, I am not sure if there is a general formula like there is for derivative. There are many rules, such as the power rule. The power rule states that:
$$\int x^n \ dx =\dfrac{x^{n+1}}{n+1}+C$$
So the integral
$$\int x^3 \ dx$$
would be equal to:
$$\dfrac{x^{3+1}}{3+1}+C$$
$$=\dfrac{x^4}{4}+C$$
Some important integral rules are:
$$\int a \ dx = ax+C \ \ \text{($a$ is a constant)}$$
$$\int cf(x) \ dx = c\int f(x) \ dx \ \ \text{($c$ is a constant)}$$
$$\int f(x)g'(x) \ dx = f(x)g(x)-\int g(x)f'(x) \ dx \ \ \text{(this is called Integration By Parts)}$$
$$\int x^n \ dx = \dfrac{x^{n+1}}{n+1} + C, \ \ \ n\neq -1 \ \ \text{(Power rule)}$$
$$\int (f\pm g)(x) \ dx = \int f(x) \ dx \pm \int g(x) \ dx \ \ \text{(Sum and Difference rule)}$$
$$\int \dfrac{1}{x} \ dx = \ln|x|+C \ \ \text{(Reciprocal rule)}$$
$$\int \dfrac{1}{x^2+a^2} \ dx = \dfrac{1}{a}\tan^{-1}\left(\dfrac{x}{a}\right)+C \ \ \text{(Reciprocal of sum of squares. $a$ is a constant)}$$
$$\int \dfrac{1}{x^2-a^2} \ dx = \dfrac{1}{2a}\ln\left|\dfrac{x-a}{x+a}\right|+C \ \ \text{(Reciprocal of difference of squares. $a$ is a constant)}$$
$$\int e^x \ dx = e^x + C \ \ \text{(The antiderivative is the same as the integrand)}$$
$$\int a^x \ dx = \dfrac{a^x}{\ln a} + C \ \ \text{($a$ is a constant)}$$
There are many more integration rules. Look them up online if you wish. Click here to access a list of integration rules.
Thank you for reading. I hope that I have helped you.
