Math 166 - Study Guide: Exam #2
Integration: Which method do I use?
1. Try substitution first:
• Might need to manipulate the integrand; i.e. Cleverly multiply by one or add 0.
• Number on top with quadratic on bottom? Use completing the square to get tan−1 ().
2. Try integration by parts:
• What should I chose for u? Look at my notes for ILATE for educated guess.
• Keep in mind that u goes down (derivative) and dv goes up (integral).
3. If it is a rational expression, try Partial Fraction Decomposition:
• If the denominator is linear, then the numerator is A.
• If the denominator is quadratic, then the numerator is Ax + B.
• You need a term for every power of everything in the denominator.
Example:
4x3 − 5x + 1
A
Cx + D
B
Ex + F
= + 2+ 2
+ Gx + H(x2 + 3)3
+ 2
2
2
3
x (x + 3)
x
x
x +3
(x + 3)2
4. Is it a trig function to a power? i.e. sinn (x), cosm (x), sinn (x) cosn (x):
• If both powers are even, then use the double angle formulas to get all first order terms.
• If at least one power is odd, then break off one from the odd power(s) and apply the Pythagorean
Identity (sin2 (x) + cos2 (x) = 1).
5. Sine function times Cosine function:
• If you have sin(nx) cos(mx), n 6= m, use the trig product formulas.
6. What to do when Substitution and Int by Parts doesn’t work on integrands with radicals:
√
• Linear term under radical ( n ax + b):
Try u = (ax + b)1/n and then solve for x and take the derivate to calculate dx.
√
√
√
• Quadratic term under the radical ( n a2 − x2 , n x2 − a2 , n a2 + x2 ):
Use the fact that sin2 (x) + cos2 (x) = 1 and 1 + tan2 (x) = sec2 (x) to determine the appropreiate
substitution.√
Example: a2 − x2 with 1 − sin2 (x) = cos2 (x) and x = a sin(t). Hence,
q
q
p
p
2
2
2
2
2
a − x = a − a sin (t) = a2 (1 − sin2 (t)) = a cos2 (t) = a cos(t)
7. Other Notes:
• I will expect that you know the integrals/derivatives of the basic (polynomial, rational, logarithmic, exponential, roots, trig, and inverse trig) functions.
• You will NOT be given the identities sin2 (x) + cos2 (x) = 1 and 1 + tan2 (x) = sec2 (x), but I will
give you the other trig identites discussed in class.
• I will also give you any necessary integrals if I view it as a nonbasic integral, or I will specify
in the directions to only set up the integral. Thus, it is very important that you READ THE
DIRECTIONS for every question.
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