Integration and the Fundamental Theorem

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$)

$\int \frac{1}{x}\,dx = \ln|x| + C$, $\int e^x\,dx = e^x + C$

$\int \sin x\,dx = -\cos x + C$, $\int \cos x\,dx = \sin x + C$

$\int_a^b f(x)\,dx = F(b) - F(a)$

where $F'(x) = f(x)$.

$\text{Area} = \int_a^b |f(x)|\,dx$. Split at roots if $f(x)$ changes sign.

$\int_1^3 (2x + 1)\,dx = [x^2 + x]_1^3 = (9+3) - (1+1) = 10$.