AP Calc AB Unit 4

According to the FTC, the derivative of an antiderivative is equal to…

the original function

∫du/(a²+u²)

(1/a)arctan(u/a)+C

∫du/(u²-a²)

(1/2a)ln|(u-a)/(u+a)|+C

∫du/√(a²-u²)

arcsin(u/a)+C

average value formula

(1/(b-a))∫f(x)dx

∫(1/x)dx

ln x + C

∫e^x dx

e^x + C

∫sin x dx

-cos x + C

∫cos x dx

sin x + C

∫sex²x dx

tan x + C

∫csc²x dx

-cot x + C

∫sec x tan x dx

sec x + C

∫csc x cot x dx

-csc x + C

∫dx/√(1-x²)

arcsin x + C

∫-dx/√(1-x²)

arccos x + C

∫dx/(1+x²)

arctan x + C

∫-dx/(1+x²)

acrcot x + C

∫dx/(|x|√(x²-1))

arcsec x + C

∫-dx/(|x|√(x²-1))

arccsc x + C

LRAM area for curves that are INCREASING on the interval and area for curves that are DECREASING on the interval

underestimates; overestimates

RRAM area for curves that are INCREASING on the interval and area for curves that are DECREASING on the interval

overestimates; underestimates

area of a trapeziod

A=(1/2)(b1+b2)h

Trapezoid rule (when subintervals are equal

T=(1/2)((b-a)/n)(f(x0)+2(f(x1))…+f(xn))

how to find total area

1. find the zeros of the function

2. integrate function over each subinterval

3. add absolute values of each integrated subinterval

∫du/(a²+u²)

((1/a)arctan(u/a))/du + C

∫du/(u²-a²)

(1/(2a))(ln|(u-a)/(u+a)|)/du + C

∫du/√(a²-u²)

(arcsin(u/a))/du + C

If g(x)=∫f(x)dx, then g’(x)= and g’’(x)=

f(x); f’(x)