Green Sheet integrals

∫uⁿdu

(uⁿ⁺¹) / (n+1)

∫du/u

ln |u| + c

∫e^udu

e^u+c

∫ln u du

u ln u - u + c

∫ cos u du

sin u + C

∫ sin u du

-cos u +c

∫sec²x u du

tan u + C

∫ csc²u du

-cot u + C

∫a^udu

( a^u / ln(a) ) + c

∫ sec u tan u du

sec u + c

∫ csc u cot u du

-csc u + C

∫ (du) / (a² - u²)

sin^-1 (u/a) + c

∫ du / (a² + u²)

(1/a) tan^-1 (u/a) + c

∫tanu du

-ln |cos u| + C

∫ cot u du

ln |sin u| + c

∫ sec u du

ln |sec u + tan u| + c

∫ csc u du

-ln |csc u + cot u| + C

∫log_au du

(1/ lna) (ulnu - u) + c

carrying capacity

the number of people, other living organisms, or crops that a region can support without environmental degradation

dP/dt

kP (M-P)

P

M / 1 + Ae^-MKt

exponential change

if y changes at a rate proportional to the amount present

y=Ae^kt

exponential change

concave up

f^’’(x) greater than 0

concave down

f^’’(x) less than 0

Underestimate

second derivate is concave up

Overestimate

Second derivate concave down

Polynomial Long Divison

When the numerator is raised to a higher or equal function than denominator this must be done first