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Sept. 24, 2013 Math 2254 sec 010 Fall 2013Substitution for the form
√a2 − x2
() September 23, 2013 1 / 50
Substitution for the form√
x2 − a2
() September 23, 2013 2 / 50
Substitution for the form√
a2 + x2
We’ll assume that θ is in an appropriate interval–e.g.(−π
2 ,π2
)for the
substitution x = a tan θ() September 23, 2013 3 / 50
Evaluate The Integral
(e)∫
x5 dx√x2 + 1
() September 23, 2013 4 / 50
() September 23, 2013 5 / 50
() September 23, 2013 6 / 50
Evaluate The Integral (completing the square)
(f)∫ 9
5
dt√t2 − 6t + 13
() September 23, 2013 7 / 50
() September 23, 2013 8 / 50
() September 23, 2013 9 / 50
Section 7.4: Rational Functions, Partial Fractions
Simplify
1x − 3
− 2x + 4
() September 23, 2013 10 / 50
Now evaluate the integral∫10− x
x2 + x − 12dx
() September 23, 2013 11 / 50
We sort’a cheated! The big question is:
If we started with the simplified total fraction
−x + 10x2 + x − 12
,
how could we figure out that it decomposes into the sum of the smallerpartial fractions
1x − 3
− 2x + 4
?
() September 23, 2013 12 / 50
Rational Functions
Recall that a rational function is one of the form
f (x) =P(x)Q(x)
where P and Q are polynomials.
The function is called a proper rational function if
degree(P(x)) < degree(Q(x)).
() September 23, 2013 13 / 50
Rational Functions
If degree(P(x)) ≥ degree(Q(x)), then f is an improper rationalfunction. In this case, we can write
f (x) = p(x) +r(x)Q(x)
where p is a polynomial, and r(x)/Q(x) is proper. We can obtain thisusing long division.
dividenddivisor
= quotient +remainder
divisor
() September 23, 2013 14 / 50
Decomposing Proper Rational Functions
Theorem: Every polynomial Q(x) with real coefficients can befactored into a product
Q(x) = q1(x)q2(x) · · · qk (x)
where each qi is either a linear factor (i.e. qi(x) = ax + b) or anirreducible quadratic (i.e. qi(x) = ax2 + bx + c where b2 − 4ac < 0).
Knowing that such a factorization exists, and being able to compute itare two different animals! But at least we can know that the cases tobe outlined cover all contingencies.
() September 23, 2013 15 / 50
Decomposing Proper Rational Functions
Let f (x) = P(x)/Q(x) be a proper rational function, and let Q(x) befactored completely into linear and irreducible quadratic factors
f (x) =P(x)
q1(x)q2(x) · · · qk (x).
We’ll consider four cases(i) each factor of Q is linear and none are repeated,(ii) each factor of Q is linear and one or more is repeated,(iii) some factor(s) of Q are quadratic, but no quadratic is repeated,(iv) Q has at least one repeated quadratic factor.
() September 23, 2013 16 / 50
Case (i) Non-repeated Linear Factors
Suppose Q(x) = (a1x + b1)(a2x + b2) · · · (akx + bk ). And no pair of a’sand b’s (both) match. Then we look for a decomposition of f in the form
P(x)Q(x)
=A1
a1x + b1+
A1
a2x + b2+ · · ·+ Ak
akx + bk.
For example10− x
(x − 3)(x + 4)=
Ax − 3
+B
x + 4.
Note that each fraction in the expansion is a proper rational functionwith denominator a line.
() September 23, 2013 17 / 50
Example: Evaluate the integral∫4x − 2x3 − x
dx
() September 23, 2013 18 / 50
() September 23, 2013 19 / 50