Calculus 2 Unit 7 ( Test 2 )

Subdecks (3)

Cards (86)

  • Integration By Parts Formula
    int (udv) = uv - int(vdu)
  • ILATE (Whichever function comes first in the following list should be u)
    I : Inverse Trig
    L : Logarithmic Functions
    A : Algebraic Functions (Polynomials)
    T : Trigonometric Functions
    E : Exponential Functions
  • General rule for integration by parts
    Choose "u" so that the integral of "v" is possible AND so that the derivative of "u" is "better" or easier to work with
  • Make sure you pick dv to easily integrate!
  • Always check if you can do u-substitution before doing integration by parts.
    Also, if you are doing integration by parts more than two times in a problem, make sure not to forget to add all the integration to the final answer!
  • When you integrate ln(x), you will need to implement the dx value!
  • Reduction formula for
    sinn(x)dx , n  2\int_{ }^{ }\sin^{n}\left(x\right)dx\ ,\ n\ \ge\ 2
    -1/n sin^(n-1) x cos(x) + (n-1)/n int ( sin^(n-2) x dx )
  • sin^2 x = 2sin x cos x
  • sin^2 (x) + cos^2 (x) = 1
  • 1 + tan^2 (x) = sec^2 (x)
  • 1 + cot^2 (x) = csc^2 (x)
  • sin^2 (x) = (1/2 [ 1 - cos(2x) ])
  • cos^2 (x) = (1/2 [ 1 + cos(2x) ])
  • If the power of "sin" is odd, keep one factor of "sin" and use Pythagorean identity to change sin^2 (x) = 1 - cos^2 (x)
    If the power of "cos" is odd, do the same for "cos"
    If both are odd, pick one that will give power "2" ( if possible )
  • If both powers are even, use the half-angle identity
  • General idea is that with trig to the odd power, break it into one even and one odd in order to use the trig identities.
  • Int ( tan^n(x) sec^m(x)) dx
    Int ( cot^n(x) csc^m(x)) dx
    If power of "tan" is ODD, keep 1 factor of sec(x)tan(x) and use tan^2(x) = sec^2(x) -1
    If power of "sec" is EVEN, strip off "sec^2(x)" and use sec^2(x) = tan^2(x) + 1
    Same format for int( cot^n(x) csc^m(x))dx
  • For integrals where sine and cosine have different angles:
    sin(mx)sin(nx) = 1/2 [ cos(mx - nx) - cos(mx + nx) ]
    sin(mx)cos(nx) = 1/2 [ sin(mx - nx) + sin(mx + nx) ]
    cos(mx)cos(nx) = 1/2 [ cos(mx - nx) + cos(mx + nx) ]
  • Trigonometric Substitution, sqrt ( a^2 - x^2 )
    Side, sin(theta) = x/a -> x = a sin(theta)
    Use: 1 - sin^2(theta) = cos^2(theta)
  • Trigonometric Substitution, sqrt (a^2 + x^2)
    Hypotenuse, tan(theta) = x/a -> x= a tan(theta)
    Use: 1 + tan^2(theta) = sec^2(theta)
  • Trigonometric Substitution, sqrt (x^2 - a^2)
    Side, sec(theta) = x/a -> x= a sec(theta)
    Use: sec^2(theta) - 1 = tan^2(theta)
  • In trigonometric substitution, you have to put sides in a relationship such that we get "x/a" for "tan", "sin" or "sec"
  • a^2 + b^2 = c^2
    sqrt ( a^2 + b^2 ) = c
    a = sqrt ( b^2 - c^2 )
  • sin = opposite / hypotenuse
  • tan = opposite / adjacent
  • sec = hypotenuse / adjacent
  • cos = adjacent / hypotenuse
  • soh cah toa
  • In trigonometric substitution, don't forget to convert back from theta to terms of x!
  • (ln(x))ndx=\int_{ }^{ }(\ln(x))^{n}dx=x(ln(x))nn(ln(x))n1dxx(\ln(x))^{n}-n\int_{ }^{ }(\ln(x))^{n-1}dx
    Reduction formula with u= ((ln(x))^n
  • cos^2(x) = 1/sec^2(x)
  • cos(2x)dx = cos^2(x) - sin^2(x)
  • sin(2x) = 2 sin(x)cos(x)
  • cho (csc) , sha (sec), cao (cot)
  • ln(1) =0
  • cos(pi/2) = 0
    sin(pi/2) = 1
  • cos(0) = 1
    sin(0) = 0
  • Trapezoidal Rule
    (b - a)/2n
    pattern = 1, 2, 2, 2, ... 2, 1
  • delta x = (b -a) /n
  • Midpoint Rule
    Involves finding the average
    x̄ = 1/2 ( x2 - x1 )
    ( b - a)/n