AP Calculus AB/BC - Unit 6: Integration and Accumulation of Change

$\int du = u + C$

$\int u^n du = \frac{u^{n+1}}{n+1} + C, \quad n \neq -1$

$\int \frac{du}{u} = \ln |u| + C$

$\int e^u du = e^u + C$

$\int a^u du = \frac{a^u}{\ln a} + C$

$\int \sin u \, du = -\cos u + C$

$\int \cos u \, du = \sin u + C$

$\int \tan u \, du = -\ln |\cos u| + C$

$\int \cot u \, du = \ln |\sin u| + C$

$\int \sec u \, du = \ln |\sec u + \tan u| + C$

$\int \csc u \, du = -\ln |\csc u + \cot u| + C$

$\int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1} \frac{u}{a} + C$

$\int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1} \frac{u}{a} + C$

$\int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a} \sec^{-1} \left| \frac{u}{a} \right| + C$

A **Riemann Sum** is a rectangular approximation method where you DO NOT EVALUATE THE INTEGRAL, but sum the areas of the rectangles.

For uneven intervals, calculate one trapezoid at a time:

$A_{\text{trap}} = \frac{1}{2} h (b_1 + b_2)$

For even intervals:

$$\int_a^b f(x) dx = \frac{b-a}{2n} \left[ y_0 + 2y_1 + 2y_2 + \dots + 2y_{n-1} + y_n \right]$$

If $f(x)$ is continuous on $[a, b]$, then there exists a number $x = c$ such that:

$$f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)dx$$

This value $f(c)$ represents the **average value** of the function on $[a, b]$.

$x(t)$ = position function

$v(t)$ = velocity function

$a(t)$ = acceleration function

The derivative of position is velocity: $v(t) = x'(t)$.

The derivative of velocity is acceleration: $a(t) = v'(t)$.

Displacement:

$\int_{t_0}^{t_f} v(t) dt$

Distance:

$\int_{\text{initial time}}^{\text{final time}} |v(t)| dt$

Average Velocity:

$$\text{Average Velocity} = \frac{\text{Final position} - \text{Initial position}}{\text{Total time}} = \frac{\Delta x}{\Delta t}$$

The total amount $F(x)$ at any time $x$ is given by:

$$F(x) = f(a) + \int_a^x f'(t) dt$$

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

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

$$\frac{d}{dx} \int_a^{g(u)} f(t) dt = f(g(u)) \frac{du}{dx}$$