Chapter 8-TECHNIQUES OF INTEGRATION
Text book :Calculus by
Thomas, 11th edition.
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8.1 Basic Integration Formulas
In this chapter we study a
number of important
techniques for finding
indefinite integrals of more
complicated functions than
those seen before
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EXAMPLE 1: Evaluate
Making a Simplifying Substitution
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EXAMPLE 2
Evaluate
Completing the Square
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EXAMPLE 3 Evaluate
Eliminating a Square Root
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Reducing an Improper Fraction
EXAMPLE 4 Evaluate
degree of numerator greater than or equal to degree of denominator
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EXAMPLE 6 Evaluate
Separating a Fraction
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H.W 8.1
Evaluate each integral in Exercises 1–29 by using a substitution to reduce
it to standard form.
Evaluate each integral in Exercises 47–50 by reducing the improper fraction and using a substitution (if necessary)
to reduce it to standard
form.
Evaluate the integral by separating the fraction
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8.2 Integration by Parts
EXAMPLE 1 Using Integration by Parts Find
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EXAMPLE 3 Integral of the Natural Logarithm Find
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EXAMPLE 4 Repeated Use of Integration by Parts
Evaluate
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EXAMPLE 5 Evaluate
Solving for the Unknown Integral
The unknown integral now appears on both sides of the equation. Adding the integral to both sides and adding the constant of integration gives
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Evaluating Definite Integrals by Parts
EXAMPLE 6 Find the area of the region bounded by the curve
and the x-axis from x=0 to x=4
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Tabular Integration
EXAMPLE 7 Using Tabular Integration
Evaluate
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EXAMPLE 8 Using Tabular Integration
Evaluate
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EXAMPLE 9 A Reduction Formula
When n is a positive integer
EXAMPLE 10 Using a Reduction Formula
Evaluate
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Example Finding area Find the area of the region enclosed by
the curve
𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑢 = 𝑥 𝑑𝑢 = 𝑑𝑥 𝑑𝑣 = 𝑠𝑖𝑛𝑥 𝑑𝑥, 𝑣 = − cos 𝑥
𝜋
𝜋
Area 𝑆1 = ‫׬‬0 𝑥 sin 𝑥 𝑑𝑥 = ሾ−𝑥 𝑐𝑜𝑠𝑥 ȁ 𝜋0 + ‫׬‬0 cos 𝑥 𝑑𝑥 = 𝜋 + 𝑠𝑖𝑛𝑥 𝜋0 = 𝜋
H.W. 8.2 Evaluate the integrals by part
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8.3 Integration of Rational Functions by Partial Fractions
න
5𝑥 − 3
𝑑𝑥
𝑥 2 − 2𝑥 − 3
5𝑥−3
5𝑥−3
=
𝑥 2 −2𝑥−3
𝑥+1)(𝑥−3
𝐴
𝐵
=
+
𝑥+1
𝑥−3
1
A and B undetermined coefficients until proper values for them have been found
Multiply both sides of 1 by (x+1)(x-3)
the coefficients of like powers of x on the two sides are equal:
Solving these equations simultaneously gives
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EXAMPLE 1 Distinct Linear Factors
Evaluate
using partial fractions
Multiply both sides by (x-1)((x+1)(x+3)
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equate coefficients of like powers of x obtaining
1
2
3
add 1 and 3
Add 2 and 4
8𝐴 = 6
substitute A in 4
4𝐴 − 2𝐵 = 2
4
𝐴 = 3/4
B=1/2, sub. A and B in 1
c=-1/4
where K is the arbitrary constant of integration
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EXAMPLE 2 A Repeated Linear Factor
Evaluate
Solution First we express the integrand as a sum of partial fractions with undetermined coefficients
Equating coefficients of corresponding powers of x gives
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EXAMPLE 3 Integrating an Improper Fraction
Evaluate
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EXAMPLE 6 Using the Heaviside Method
Find A, B, and C in the partial-fraction expansion
Solution If we multiply both sides of Equation (3) by(x-1) to get
and set x=1 the resulting equation gives the value of A:
covered the factor (x-1) in the denominator of the original fraction
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and evaluated the rest at x=1
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H.W integrate the following
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Trigonometric Integrals
Products of Powers of Sines and Cosines
න𝑐𝑜𝑠 𝑚 𝑥 𝑐𝑜𝑠 𝑛 𝑥 𝑑𝑥
where m and n are nonnegative integers (positive or zero). We can divide the work into three cases
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Example
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