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


The Product Rule in Reverse


6.1 Integration by Parts


Some Examples

EXAMPLE: Calculate x cos (x) dx.

EXAMPLE: Calculate ln(x) dx.




Advanced Examples

EXAMPLE: Calculate the integral




Reduction Formulas

EXAMPLE: Let a be a nonzero constant. Derive the reduction formula

EXAMPLE: Evaluate x3e-x dx




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


Squares of Sine, Cosine, Secant, and Tangent

EXAMPLE: Show that

and


6.2 Powers and Products of Trigonometric Functions


Higher Powers of Sine, Cosine, Secant, and Tangent




Higher Powers of Sine, Cosine, Secant, and Tangent

EXAMPLE: Derive the formula

EXAMPLE: Evaluate cos6(x) dx.




Odd Powers of Sine and Cosine

EXAMPLE: Evaluate sin5(x) dx.

EXAMPLE: Evaluate cos3(x) dx.




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.




Integrals which Involve Both Sine and Cosine

EXAMPLE: Evaluate cos3(x) sin4(x) dx.

EXAMPLE: Evaluate sin4(x) cos6(x)dx.




Converting to Sines and Cosines

EXAMPLE: Evaluate tan5(x) sec3(x) dx.




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


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


6.3 Trigonometric Substitution


EXAMPLE: Calculate

EXAMPLE: Calculate




EXAMPLE: Calculate

EXAMPLE: Calculate

General Quadratic Expressions that Appear Under a Radical




EXAMPLE: Calculate

Quadratic Expressions Not Under a Radical




1. What indirect substitution is appropriate for




Quick Quiz

Chapter 6-Techniques of Integration6.4 Partial Fractions-Linear

Factors


The Method of Partial Fractions for Linear Factors

EXAMPLE: Integrate


6.4 Partial Fractions-Linear Factors


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.




The Method of Partial Fractions for Distinct Linear Factors

EXAMPLE: Calculate the integral




Repeated Linear Factors

For each aj:




Repeated Linear Factors

EXAMPLE: Evaluate the integral




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


Rational Functions with Quadratic Terms in the Denominator


6.5 Partial Fractions-Irreducible Quadratic Factors


Rational Functions with Quadratic Terms in the Denominator

EXAMPLE: State the form of the partial fraction decomposition of




Checking for Irreducibility

EXAMPLE: Find the correct form of the partial fraction decomposition for the rational expression




Calculating the Coefficients of a Partial Fractions Decomposition

EXAMPLE: Find the partial fraction decomposition of the rational function 3/(x3 + 1)

EXAMPLE: Calculate




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



Chapter 6-Techniques of Integration6.6 Improper Integrals-

Unbounded Integrands


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.


6.6 Improper Integrals-Unbounded Integrands


Integrals with Infinite Integrands


EXAMPLE: Analyze the integral





Integrals with Interior Singularities





Integrands That Are Unbounded at Both Ends

EXAMPLE: Determine whether the improper integral below converges or diverges.




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−.




Proving Convergence Without Evaluation

EXAMPLE: Show the following improper integral is convergent




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


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.


6.7 Improper Integrals-Unbounded Intervals


The Integral on an Infinite Interval

EXAMPLE: Calculate the improper integral

EXAMPLE: Determine whether the following improper integral converges or diverges.




Integrals over the Entire Real Line

EXAMPLE: Determine whether the improper integral below converges or diverges.

EXAMPLE: Evaluate the improper integral




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.




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.




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:

Chapter 6-Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley &...

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Transcript of Chapter 6-Techniques of Integration Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley &...