Summary of Fundamental Theorem of Calculus, Integrals, Average Value, and Riemann Sum

Fundamental Theorem of Calculus

If we integrate a function f(x), then \int_{a}^{b} f(x) dx = g(b) - g(a), where g is an antiderivative of f (i.e., g'(x) = f(x)).
Alternatively, \int_{a}^{b} g'(x) dx = g(b) - g(a).
For definite integrals, if \int{0}^{x} f(t) dt, then the derivative is \frac{d}{dx} \int{0}^{x} f(t) dt = f(x).
More generally, \frac{d}{dx} \int_{a}^{x} f(t) dt = f(x).

Power function: \int x^n dx = \frac{1}{n+1}x^{n+1} + C if n \neq -1.
If n = -1, then \int \frac{1}{x} dx = \ln|x| + C.
Integration is a linear operation.
- \int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx
- \int k \cdot f(x) dx = k \int f(x) dx
Exponential function: \int e^{kx} dx = \frac{1}{k}e^{kx} + C.
Generalized power rule: \int u^n du = \frac{1}{n+1}u^{n+1} + C if n \neq -1.
For non-integer exponents: \int x^\alpha dx = \frac{1}{\alpha + 1}x^{\alpha + 1} + C if \alpha \neq -1.
\int a e^{kt} dt = a \cdot \frac{1}{k} e^{kt} + C.
\int 10^x dx = \frac{1}{\ln 10} 10^x + C. More generally, \int a^x dx = \frac{1}{\ln a} a^x + C (for a > 0 and a \neq 1).

The average of a function f(x) over an interval [a, b] is given by \frac{1}{b-a} \int_{a}^{b} f(x) dx.

The definite integral is equal to the limit of the Riemann sum: \int{a}^{b} f(x) dx = \lim{n \to \infty} [\text{Riemann Sum}].
Divide the interval [a, b] into n equal parts, each of length \frac{b-a}{n}.
Left Endpoint Riemann Sum: \sum_{k=0}^{n-1} f(a + k \frac{b-a}{n}) \cdot \frac{b-a}{n}.
Right Endpoint Riemann Sum: \sum_{k=1}^{n} f(a + k \frac{b-a}{n}) \cdot \frac{b-a}{n}.
If not specified, Riemann Sum refers to right endpoint Riemann sum.