AP Calculus AB Formulas

(sin x)'

cosx

(cos x)'

-sinx

(tan x)'

sec^2 x

(sec x)'

secx tanx

(csc x)'

-cscx cotx

(cot x)'

-csc^2 x

(arcSin u)'

u' / (1-u^2)^1/2

(arcCos x)'

- u' / (1- u^2)^1/2

(arcTan x)'

u' / u^2 + 1

(arcSec x)'

u' / |u| (u^2 - 1)^1/2

(arcCsc x)'

- u' / |u| (u^2 - 1)^1/2

(arcCot x)'

- u' / u^2 + 1

(Ln u)'

u' / u

(e^(u))'

u' e^(u)

ʃ x^n dx

(x^(n+1)) / (n+1) + C

ʃ 1/x dx

Ln x + C

ʃ 1/ x^2 dx

-1/x + C

ʃ dx /(x + b) dx

Ln |x + b| + C

ʃ sinx dx

-cosx + C

ʃ sin ax dx

(-cos ax) / a + C

ʃ cosx dx

sinx + C

ʃ cos ax dx

(sin ax) / a + C

ʃ tanx dx

{ Ln |secx| + C

- Ln |cosx + C }

ʃ secx dx

Ln |secx + tanx| + C

ʃ secx tanx dx

secx + C

ʃ sec^2 x dx

tanx + C

ʃ e^x dx

e^x + C

ʃ e^(ax) dx

(1/a) e^(ax) + C

ʃ dx /(x + b)

Ln |x + b| + C

ʃ dx /(ax + b)

(1/a) Ln |ax + b| + C

First Fundamental Theorem

ʃab f(x) dx = F(b) - F(a)

ʃab f '(x) dx = f(b) - f(a)

* f is continuous on (a, b)

* F is any anti-derivative of f

Second Fundamental Theorem

F(x) = ʃax f(t) dt

F'(x) = f(x) or d/dx (ʃax f(t) dt) = f(x)

* f is continuous on the open interval

Trapezoid Rule

ʃab f(x) dx ~= ((b-a)/2n) [A + 2B + 2C + 2D + E] if n=4

Right RAM

(w1 B) + (w2 C) + (w3 D) + (w4 E) [A,E]

w = distance from previous x point, [letter] = y value at the point

Left RAM

(w1 A) + (w2 B) + (w3 C) + (w4 D) [A,E]

w = distance from previous x point, [letter] = y value at the point

Midpoint RAM

(w1 A) + (w2 B) + (w3 C) + (w4 D) + (w4 * D) [A,E]

(use the x and y coordinates in the MIDDLE of each region)