Studied by 1 personAccording 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
find the zeros of the function
integrate function over each subinterval
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)
Note
Flashcard