trigonometric identities

sin²+cos²

1

integral of 1/(1-x) dx

-ln|1-x|

integral of sec x dx

ln |sec x + tan x| + C

sec² -tan²

1

csc²-cot²

1

sin²

(1-cos(2x)/2

cos²

(1+cos(2x))/2

sin(2x)

2sinxcosx

cos(2x) - three answers

cos²x-sin²x

2cos²x-1

1-2sin²x

tan(2x)

(2tanx)/(1-tan²x)

sec²

1+ tan²

sin(x)cos(x)

½ sin (2x)

3 ways of thinking about the definite integral

1. net change (if f is a rate of change)

2. area between f & the horiz. axis

3. limit of Riemann sums

FTOC I & II

1. if f is a cont. function, a is any number, and we let A_f(x) = integral from a to x of f(t) dt, the d/dx A_f(x)= f(x). take derivative WRT x of a function of x so result is a function of x.

2. if f(x) is a continuous function on [a,b] and F is an antiderivative of f, that is, F’=f, then the integral from a to b of f(t) dt = F(b) - F(a)

integral of xⁿ dx

(xⁿ⁺¹)/(n+1)+C

integral of 1/x dx

ln|x| +C

integral of e^x dx

e^x+C

integral of b^x when b is a positive constant

b^x/ln b + C

integral of 1/x² dx

-1/t +C

integral of sin x dx

-cosx + C

integral of secxtanx dx

secx + c

integral of cscxcotx dx

-cscx + C

integral of cosx dx

sin x + C

integral of 1/(1+x²) dx

arctan x + C

integral of csc²x dx

-cot x + C

integral of sec²x dx

tan x + C

integral of 1/sqrt(1-x²)

-arccos x + c = arcsin x + C