### Symbol
The sum symbol $\Sigma$ is replaced in the limit by the integral symbol $\int$ , whose origin is in the letter "S". Think of the integral sign as a "Super S;" that is, think of it as a Super Sum of Small areas under a curve.
$\int$
### General Rules of Integration
#### Constant Rule
$\int c \ dx = c \ \int dx \ = cx + C$
Where c is a constant and C is the constant of integration. This is just a specific case of the Power Rule.
#### Power Rule
Add 1 to the exponent, then divide by the new exponent.
$\int x^n \ dx = \frac{x^{n+1}}{n+1} + \ C$
$\int x \ dx = \frac{1}{2}x^2$
#### Natural Logarithm Rule
$\int \frac{1}{x}dx = \ln |x| + C \quad \quad \quad (\text {for} \ x \neq 0)$
#### Exponential Rule
$\int e^x \ dx \ = \ e^x + C$
#### Basic Trigonometric Integrals
$\int sin(x) \ dx \ = \ -cos(x) \ + \ C $
$\int cos(x) \ dx \ = \ sin(x) \ + \ C $
$\int sec^2(x) \ dx \ = \ tan(x) \ + \ C $
$\int csc^2(x) \ dx \ = \ -cot(x) \ + \ C $
### Definite Integral
$\int ^{a}_{b} f(x) \ dx$
Read as "the integral from $a$ to $b$ of $f$ of $x$ dee $x
quot;
![[Pasted image 20241009093946.png]]
We can also define the following:
$\int ^{a}_{b} f(x) \ dx = - \int ^{b}_{a} f(x) \ dx$
And
$\int ^{a}_{a} f(x) \ dx = 0$
### Sample Problem
$\int _3^5 x \ dx = \frac{1}{2}x^2 \ \Biggr| _3^5 \tag{1}$
$\int _3^5 x \ dx = \ \Biggr(\frac{1}{2} \cdot (5^2) \Biggr) - \Biggr(\frac{1}{2} \cdot (3^2) \Biggr) \tag{2}$
$\int _3^5 x \ dx = \frac{25}{2} - \frac{9}{2} \tag{3}$
$\boxed {\int _3^5 x \ dx = 8} \tag{4}$