Calculus and Analytical Geometry 2 - Vectors, Improper Integrals, IBP, and Trig Sub

dot product = negative

obtuse angle

dot product = positive

acute angle

dot product = 0

perpendicular/orthogonal

unit vector in same direction as a

a/|a|

projection of a onto b

projb(a)=((a*b)/|b|)*(b/|b|)

component of a onto b

compb(a)=(a*b)/|b|

dot product with magnitudes and angles

|a|*|b|*cos(ø) where ø is the difference between the two angles

Volume of a parallelepiped

|a * (b x c)|

cross product

matrix

area of a parallelogram

|axb|

find a vector orthogonal to u and v

uxv or |vxu|

int(f(kx)dx)

(1/k)(F(kx))+C

d/dx (sqrt(x))

1/2sqrt(x)

what should IBP be used for

integrals of inverse functions (ln and inverse trig)

PFD for non-repeated linear factors

(something)/(x+c)(x+d)=A/(x+c)+B/(x+d)

PFD - repeated linear factors

something/(x+c)^2=A(x+c)+B(x+c)^2

3 Pythagorean Identities

sin^2x+cos^2x=1
tan^2x+1=sec^2x
1+cot^2x=csc^2x

Double Angle Formulas

sin(2x)=2sin(x)cos(x)
cos(2x)=cos^2-sin^2=1-2sin^2=2cos^2-1

int(tanx)

ln|secx|

int(cotx)

ln|sinx|+c

int(secx)

ln|secx+tanx|+C

int(cscx)

-ln|cscx+cotx|+c

int(sin^5xcos^4x)

u=cosx

a^2-x^2

x=asin(theta)

a^2+x^2

x=atan(theta)

x^2-a^2

x=asec(theta)

int(2cos^2(3x)sin^3(3x)dx

-cos^4(3x)/4+cos^5(3x)/5

even powers only

double angle identity

convergent

improper integral exists and is finite

divergent

improper integral dne, area under curve infinite

lim(x->inf)(polynomial/polynomial) and degree top>bottom

limit is inf

lim(x->inf)(polynomial/polynomial) and degree top

limit is 0

lim(x->inf)(polynomial/polynomial) and degree top=bottom

limit is ratio of leading coefficients

lim(x->inf)lnx

inf

lim(x->0^+)lnx

-inf

lim(x->inf)e^x

inf

lim(x->-inf) e^x

0

if denominator has 6 terms (1 repeated 2x and 1 repeated 3x), how many fraction terms will pfd have

6

if denominator is x^2+4, numberator is

Ax+B

when you do u sub with powers of sine and cosine, how can you tell whether u should be sine or cosine

u should be the one with even power

how do you know when to use double angle identities

even-even

how do you know when to use pythagorean identities

odd-even or odd-odd

what does the dot product do for us?

how similar two vectors are in their direction