Chapter 6-Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

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 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  x 3 e -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 Quiz 1. 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 Integration 6.2 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  cos 6 (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  sin 5 (x) dx. EXAMPLE: Evaluate  cos 3 (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 cos 2 (x) + sin 2 (x) = 1 to convert the integrand to a sum 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  cos 3 (x) sin 4 (x) dx. EXAMPLE: Evaluate  sin 4 (x) cos 6 (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  tan 5 (x) sec 3 (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 Quiz 1. What trigonometric identity is used to evaluate  sin 2 (x) dx? 2. Evaluate 3. The equation Results from what substitution 4. For what value c is (Do not integrate!)

Chapter 6-Techniques of Integration 6.3 Trigonometric Substitution Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved ExpressionSubstitution a 2 -x 2 x = a sin  dx = a cos (  ) d  a2+x2a2+x2 x = a tan  dx = a sec 2 (  ) d  x 2 -a 2 x = a sec  dx = 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

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 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 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 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 Factors To integrate a function of the form where p(x) is a polynomial of degree less than K and the a j are distinct real numbers decompose the function into the form and solve for the numerators A 1, A 2,..., A K. The result is called the partial fraction decomposition of the original 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 a j :

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 Quiz 1. True or false: If q (x) is a polynomial of degree n that factors into linear 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. For what values of A, B, and C is A(x + 1) (x + 2)+Bx (x + 2)+Cx (x + 1) = 4x 2 +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 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

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/(x 3 + 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 Quiz 1. 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 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 DEFINITION: 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 Evaluation THEOREM: (Comparison Theorem for Unbounded Integrands) Suppose that f and g are continuous functions on 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 Quiz 1. 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 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 DEFINITION: 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 Evaluation THEOREM: (Comparison Theorem for Integrals over Unbounded Intervals) Suppose that f and g are continuous functions 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 continuous functions 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 Quiz 1. 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: