Calculus 2 Unit 7 ( Test 2 )

• 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

  $\int_{ }^{ }\sin^{n}\left(x\right)dx\ ,\ n\ \ge\ 2$

  -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!
• $\int_{ }^{ }(\ln(x))^{n}dx=$$x(\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