Unit 4 Flashcards
POPULAR INTEGRALS
∫1/x dx =
ln|x| + c
∫a^x dx =
a^x / lna + c
speed =
|velocity|
an object is SPEEDING UP when
v(t) and a(t) have the same signs
an object is SLOWING DOWN when
v(t) and a(t) have opposite signs
left Riemann sum
n-1
∑ b-a/n * f(a + b-a/n i)
i=0
width * height
- right is: n
∑
i=1
midpoint Riemann sum (table)
- find difference across three x values (width)
- multiply by midpoint y value (height)
- width * height
trapezoid Riemann sum (table)
1/2 * width * b1+b2 (y values of both points)
formal definition of an integral (exact value of area under a curve)
. n-1
lim (n-> ∞) ∑ b-a/n * f(a + b-a/n i)
I=0
a = ∫f(x) dx b
fundamental theorem of calculus
a
∫f(x) dx = F(b) - F(a)
b
average value on [a, b]
. a
1/b-a * ∫f(x) dx
b
. g(t)
d/dx ∫f(t) dt =
a
f[g(x)] * g’(x)
- a is just a constant so it goes away
displacement
final - initial
a
∫v(t) dt
b
total distance
a
∫|v(t)| dt
b
TRIGONOMETRIC INTEGRALS
∫tanx dx =
-ln|cosx| + C
∫cotx dx =
ln|sinx| + C
∫secx dx =
ln|secx + tanx| + C
∫cscx dx =
-ln|cscx + cotx| + C
INVERSE TRIGONOMETRIC INTEGRALS
a = constant
u = variable
∫ 1 / (a^2 - u^2)^1/2 du =
arcsin u/a + C
∫ 1 / a^2+u^2 du =
1/a arctan u/a + C
∫ 1 / u(u^2 - a^2)^1/2 du =
1/a arcsec |u|/a + C
