Sept. 24, 2013 Math 2254 sec 010 Fall 2013Substitution for the form

√a2 − x2

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Substitution for the form√

x2 − a2

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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

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Evaluate The Integral (completing the square)

(f)∫ 9

5

dt√t2 − 6t + 13

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Section 7.4: Rational Functions, Partial Fractions

Simplify

1x − 3

− 2x + 4

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Now evaluate the integral∫10− x

x2 + x − 12dx

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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

?

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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)).

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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

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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.

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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.

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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.

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Example: Evaluate the integral∫4x − 2x3 − x

dx

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