Exam 2 Flashcards
integral(sin^m x cos^n x) dx
1. when m = odd, u = ?
2. when n=odd, u = ?
3. both even
- u=cosx
- u=sinx
- double angle formula
when you see an odd power…
take one out, ex. sec^3xtanx = sec^2xtanx^secx
integral(tan^msec^n)dx
cases
- n>0, n=even, u=tanx
- n>0, m=odd, u=secx
- n>0, m=even, n=odd, integration by parts
- n=0, tan^2(x)=sec^2(x)-1
* distribute then make u=tanx
integral of tanxdx
ln|secx|+C
integral of secx
ln|secx+tanx|
don’t forget
- change the bounds for trig-sub and u-sub
- check to see if you need to do long division before partial fractions
- rationalize
- for integration by parts, oftentimes, u=tanx and dv=tanxsecxdc
- for integration by parts, part of the problem because the integral from the beginning, so set the entire thing equal to the og integral, 2 of the same intgral in equation now, so get them on their own side, and then divide by 2 to solve for the integral :)
- numerator divided by denominator for long division, not the other way
prove area of a circle=pir^2
- equation of a circle is x^2+y^2=r^2
- get y by itself: y=+-root(r^2-x^2)
- top half of the circle is positive root, bottom half of circle is negative
- integral (bounds are -r and 4, top function minus bottom)
- solve integral using trig subsitution
how to check if a factor is indecomposable
compete the square, if sign of the square is positive, then it is indecomposable
what to do when integral of a polynomial under a root
- complete the square
- u-sub
- trig substitution
partial fractions
- check if you need to do long division (you do if the degree in numerator is greater than or equal to degree in denominator)
- if you need to do long division, rewrite integral as (long division answer*what you divided by+remainder)/(of denominator), then split the fraction to make one of the integrals 1 and write 2 integrals)
- write out partial fraction form (using A, B, C, D, etc.)
* if you can factor, than factor it - If only linear functions…
- don’t need to multiply both side by denominator when it’s only linear functions as factors ex. (x-1) or (x+5)
- set both denominator equal to 0 to solve for x
- set numerator and denominator equal to each other and solve for x (for the factor that equals 0, make it the variable) - if there is a nonlinear factor, multiply both side of equation by denom., distribute, find letters through systems of equations
know derivative of sin inverse, cos inverse, and tan inverse
how to know when to do u-sub for partial fractions
when numerator is linear and denominator is quadratic
* plugging du in so change du to equal the numerator, not changing the numerator to equal du
* completing the square sometimes gives 1/x^2+1, inverse derivative of tangent is that
* if you have du+a constant then split the integral so that du and a are split
tan^-1(infinity)
pi/2
sin^-1(infinity)
DNE
telescoping series
- rewrite problem as sigma from k=1 to infinity of function
- function = ak
- partial fractions, then factor out numerator if needed
- write out a1-a4 and an
- cancel out terms
- sn= terms remaining
- limit as n approaches infinity of sn
derivative of cos
NEGATIVE sin
-e^0
-1
e^-0
1
