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Transcript of Chapter 6-Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley &...
Chapter 6-Techniques of Integration
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Chapter 6-Techniques of Integration6.1 Integration by Parts
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Product Rule in Reverse
Chapter 6-Techniques of Integration
6.1 Integration by Parts
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Some Examples
EXAMPLE: Calculate x cos (x) dx.
EXAMPLE: Calculate ln(x) dx.
Chapter 6-Techniques of Integration
6.1 Integration by Parts
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Advanced Examples
EXAMPLE: Calculate the integral
Chapter 6-Techniques of Integration
6.1 Integration by Parts
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Reduction Formulas
EXAMPLE: Let a be a nonzero constant. Derive the reduction formula
EXAMPLE: Evaluate x3e-x dx
Chapter 6-Techniques of Integration
6.1 Integration by Parts
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz1. The integration by parts formula is another way of looking at what formula for the derivative?2. An application of integration by parts leads to an equation of the form
What are A and B?3. An application of integration by parts leads to an equation of the form
What is (x)?
Chapter 6-Techniques of Integration6.2 Powers and Products of
Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Squares of Sine, Cosine, Secant, and Tangent
EXAMPLE: Show that
and
Chapter 6-Techniques of Integration
6.2 Powers and Products of Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Higher Powers of Sine, Cosine, Secant, and Tangent
Chapter 6-Techniques of Integration
6.2 Powers and Products of Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Higher Powers of Sine, Cosine, Secant, and Tangent
EXAMPLE: Derive the formula
EXAMPLE: Evaluate cos6(x) dx.
Chapter 6-Techniques of Integration
6.2 Powers and Products of Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Odd Powers of Sine and Cosine
EXAMPLE: Evaluate sin5(x) dx.
EXAMPLE: Evaluate cos3(x) dx.
Chapter 6-Techniques of Integration
6.2 Powers and Products of Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Integrals which Involve Both Sine and Cosine
If at least one of m or n is odd, then we apply the odd-power technique If both m and n are even, then we use the identity cos2 (x) + sin2 (x) = 1 to convert the integrand to asum of even powers of sine or of cosine.
Chapter 6-Techniques of Integration
6.2 Powers and Products of Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Integrals which Involve Both Sine and Cosine
EXAMPLE: Evaluate cos3(x) sin4(x) dx.
EXAMPLE: Evaluate sin4(x) cos6(x)dx.
Chapter 6-Techniques of Integration
6.2 Powers and Products of Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Converting to Sines and Cosines
EXAMPLE: Evaluate tan5(x) sec3(x) dx.
Chapter 6-Techniques of Integration
6.2 Powers and Products of Trigonometric Functions
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz1. What trigonometric identity is used to evaluate sin2(x) dx?
2. Evaluate
3. The equationResults from what substitution
4. For what value c is(Do not integrate!)
Chapter 6-Techniques of Integration6.3 Trigonometric Substitution
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Expression Substitutiona2-x2 x = a sindx = a cos () da2+x2 x = a tandx = a sec2 () dx2 -a2 x = a secdx = a sec ()
tan()d
EXAMPLE: Calculate
Chapter 6-Techniques of Integration
6.3 Trigonometric Substitution
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Calculate
EXAMPLE: Calculate
Chapter 6-Techniques of Integration
6.3 Trigonometric Substitution
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Calculate
EXAMPLE: Calculate
General Quadratic Expressions that Appear Under a Radical
Chapter 6-Techniques of Integration
6.3 Trigonometric Substitution
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
EXAMPLE: Calculate
Quadratic Expressions Not Under a Radical
Chapter 6-Techniques of Integration
6.3 Trigonometric Substitution
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
1. What indirect substitution is appropriate for
2. What indirect substitution is appropriate for
3. What indirect substitution is appropriate for
4. What indirect substitution is appropriate for
Quick Quiz
Chapter 6-Techniques of Integration6.4 Partial Fractions-Linear
Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Method of Partial Fractions for Linear Factors
EXAMPLE: Integrate
Chapter 6-Techniques of Integration
6.4 Partial Fractions-Linear Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Method of Partial Fractions for Distinct Linear FactorsTo integrate a function of the form
where p(x) is a polynomial of degree less than K and the aj are distinct real numbers decompose the function into the form
and solve for the numerators A1, A2, . . . , AK. The result is called the partial fraction decomposition of theoriginal rational function.
Chapter 6-Techniques of Integration
6.4 Partial Fractions-Linear Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Method of Partial Fractions for Distinct Linear Factors
EXAMPLE: Calculate the integral
Chapter 6-Techniques of Integration
6.4 Partial Fractions-Linear Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Repeated Linear Factors
For each aj:
Chapter 6-Techniques of Integration
6.4 Partial Fractions-Linear Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Repeated Linear Factors
EXAMPLE: Evaluate the integral
Chapter 6-Techniques of Integration
6.4 Partial Fractions-Linear Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz1. True or false: If q (x) is a polynomial of degree n that factors into linear terms and if p (x) is a polynomial ofdegree m with m < n, then an explicitly calculated partial fraction decomposition of p (x) /q (x) requires the determination of n unknown constants.2. For what values of A, B, and C is A(x + 1) (x + 2)+Bx (x + 2)+Cx (x + 1) = 4x2 +11x+4 an identity in x?3. What is the form of the partial fraction decomposition of
4. What is the form of the partial fraction decomposition of
Chapter 6-Techniques of Integration6.5 Partial Fractions-Irreducible
Quadratic Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Rational Functions with Quadratic Terms in the Denominator
Chapter 6-Techniques of Integration
6.5 Partial Fractions-Irreducible Quadratic Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Rational Functions with Quadratic Terms in the Denominator
EXAMPLE: State the form of the partial fraction decomposition of
Chapter 6-Techniques of Integration
6.5 Partial Fractions-Irreducible Quadratic Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Checking for Irreducibility
EXAMPLE: Find the correct form of the partial fraction decomposition for the rational expression
Chapter 6-Techniques of Integration
6.5 Partial Fractions-Irreducible Quadratic Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Calculating the Coefficients of a Partial Fractions Decomposition
EXAMPLE: Find the partial fraction decomposition of the rational function 3/(x3 + 1)
EXAMPLE: Calculate
Chapter 6-Techniques of Integration
6.5 Partial Fractions-Irreducible Quadratic Factors
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz1. True or false: If q (x) is a polynomial of degree n that factors into irreducible quadratic terms and if p (x) is a polynomial of degree m with m < n, then an explicitly calculated partial fraction decomposition of p (x) /q (x) requires the determination of n unknown constants.2. What is the form of the partial fraction decomposition of
3. What is the form of the partial fraction decomposition of
4. What is the form of the partial fraction decomposition of
Chapter 6-Techniques of Integration6.6 Improper Integrals-
Unbounded Integrands
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Integrals with Infinite IntegrandsDEFINITION: If f (x) is continuous on [a, b) and unbounded as x approaches b from the left, then the value of the improper integral is defined by
provided that this limit exists and is finite. In this case we say that the improper integral converges. Otherwise the integral is said to diverge.
Chapter 6-Techniques of Integration
6.6 Improper Integrals-Unbounded Integrands
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Integrals with Infinite Integrands
EXAMPLE: Evaluate the integral
EXAMPLE: Analyze the integral
EXAMPLE: Evaluate the integral
Chapter 6-Techniques of Integration
6.6 Improper Integrals-Unbounded Integrands
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Integrals with Interior Singularities
EXAMPLE: Evaluate the integral
Chapter 6-Techniques of Integration
6.6 Improper Integrals-Unbounded Integrands
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Integrands That Are Unbounded at Both Ends
EXAMPLE: Determine whether the improper integral below converges or diverges.
Chapter 6-Techniques of Integration
6.6 Improper Integrals-Unbounded Integrands
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Proving Convergence Without EvaluationTHEOREM: (Comparison Theorem for Unbounded Integrands) Suppose that f and g are continuous functionson the interval (a, b), that 0 ≤ f (x) ≤ g (x) for all a < x < b, and that f (x) and g (x) are unbounded as x a+, or as x b−, or as x a+ and x b−.
Chapter 6-Techniques of Integration
6.6 Improper Integrals-Unbounded Integrands
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Proving Convergence Without Evaluation
EXAMPLE: Show the following improper integral is convergent
Chapter 6-Techniques of Integration
6.6 Improper Integrals-Unbounded Integrands
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz1. Calculate
2. Calculate
3. True or False: If f is unbounded at both a and b, and if c is a point in between, then is divergent if and only if both and are divergent.
4. Use the Comparison Theorem to determine which of the following improper integrals converge.
Chapter 6-Techniques of Integration6.7 Improper Integrals-
Unbounded Intervals
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Integral on an Infinite IntervalDEFINITION: Let f be a continuous function on the interval [a,∞). The improper integral is defined by
provided that the limit exists and is finite. When the limit exists, the integral is said to converge. Otherwise it is said to diverge.
Chapter 6-Techniques of Integration
6.7 Improper Integrals-Unbounded Intervals
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Integral on an Infinite Interval
EXAMPLE: Calculate the improper integral
EXAMPLE: Determine whether the following improper integral converges or diverges.
Chapter 6-Techniques of Integration
6.7 Improper Integrals-Unbounded Intervals
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Integrals over the Entire Real Line
EXAMPLE: Determine whether the improper integral below converges or diverges.
EXAMPLE: Evaluate the improper integral
Chapter 6-Techniques of Integration
6.7 Improper Integrals-Unbounded Intervals
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Proving Convergence Without EvaluationTHEOREM: (Comparison Theorem for Integrals over Unbounded Intervals) Suppose that f and g are continuousfunctions on the interval [a,1) and that 0 ≤ f (x) ≤ g (x) for all a < x.
Chapter 6-Techniques of Integration
6.7 Improper Integrals-Unbounded Intervals
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Proving Convergence Without Evaluation
EXAMPLE: Show that the following is convergent
THEOREM: (Comparison Theorem for Integrals over Unbounded Intervals) Suppose that f and g are continuousfunctions on the interval [a,1) and that 0 ≤ f (x) ≤ g (x) for all a < x.
Chapter 6-Techniques of Integration
6.7 Improper Integrals-Unbounded Intervals
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz1. Calculate
2. Calculate
3. True or false: If f is continuous on (0,1), unbounded at 0, and if c > 0, then is convergent if and only if both improper integrals and are convergent.
4. Use the Comparison Theorem to determine which of the following improper integrals converge: