Calculus with Dr. Bob V: Advanced Integration Techniques

Video Lectures

Displaying all 30 video lectures.
Lecture 1
Int by Parts 1 - Natural Log and Exponential
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Int by Parts 1 - Natural Log and Exponential
Worked problem in calculus. Calculate the indefinite integrals of: (a) x^3 ln(x) and (b) x^2 e^{-x}.
Lecture 2
Int by Parts 2 - Trig Functions
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Int by Parts 2 - Trig Functions
Worked problem in calculus. Find the indefinite integrals of the trig functions: (a) tan^{-1}(x) and (b) xsec^2(x).
Lecture 3
Int by Parts 3 - Definite Integrals
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Int by Parts 3 - Definite Integrals
Worked problem in calculus. Evaluate the definite integrals (a) int_1^e ln(x) dx and int_0^{1/4} sin^{-1}(2x) dx.
Lecture 4
Int by Parts 4 - Antiderivative of e^(2x)cos(x) (Double IBP)
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Int by Parts 4 - Antiderivative of e^(2x)cos(x) (Double IBP)
Worked problem in calculus. Using a double integration by parts, we calculate the indefinite integral of e^{2x} cos(x).
Lecture 5
Int By Parts 5 - Antiderivative for e^{3x}cos(4x) (Fast Solution)
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Int By Parts 5 - Antiderivative for e^{3x}cos(4x) (Fast Solution)
Calculus: Instead of using a double integration by parts, we give a quick solution for the antiderivative of g(x) = e^{3x}cos(4x). This method uses the definition of antiderivative and a partial formula for the antiderivative of g.
Lecture 6
Int by Parts 6 - Antiderivative of sec^3(x)
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Int by Parts 6 - Antiderivative of sec^3(x)
Worked problem in calculus. The indefinite integral of f(x) = sec^3(x) is computed. The technique is roughly 1.5 IBParts.
Lecture 7
Fast Antiderivative of x^2 exp(3x)
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Fast Antiderivative of x^2 exp(3x)
Calculus: We give a fast method for integrating x^n exp(3x) that avoids integration by parts. The two steps involve partially memorizing the answer and using the verification for antiderivatives.
Lecture 8
Integrals with cos^m(x) sin^n(x)
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Integrals with cos^m(x) sin^n(x)
Worked problem in calculus. The indefinite integrals of the form int cos^m(x) sin^n(x) dx break into two cases: one of m or n odd, or both m, n even. The first case allows for a straightforward u-substitution. The other requires the half-angle identities for cosine and sine. Examples are (a) sin^5(x), (b) sin^3(x) cos^3(x), and (c) sin^2(x) cos^2(x).
Lecture 9
Integral of cos(mx)cos(nx)
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Integral of cos(mx)cos(nx)
Worked problem in calculus. We derive the formula for the integral of cos(mx)cos(nx), and work out the special case of the definite integral int_0^2pi cos(3x)cos(2x) dx.
Lecture 10
Integral of tan^m(x) sec^n(x)
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Integral of tan^m(x) sec^n(x)
Worked problem in calculus. Two techniques are given for calculating the indefinite integral of tan^m(x) sec^n(x). We apply these to (a) sec^6(x), (b) sec^4(x) tan^2(x), and (c) tan^3(x) sec^2(x).
Lecture 11
Antiderivative of sec^5(x)
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Antiderivative of sec^5(x)
Calculus: Integration by parts is used twice to compute the antiderivative of sec^5(x), which we label as I. The method used applies to any antiderivative of sec^{odd}(x). Then we show to use the method to compute the antiderivative of sec^{odd}(x) tan^{even}(x).
Lecture 12
Integral of tan^6(x)
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Integral of tan^6(x)
Worked problem in calculus. We calculate the indefinite integral of tan^6(x), which illustrates how to compute the integral of tan^{2k} in general.
Lecture 13
Trig Substitution 1 - Basic Inverse Trig Integrals
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Trig Substitution 1 - Basic Inverse Trig Integrals
Worked problem in calculus. Basic trig substitutions are reviewed. Evaluate the indefinite integrals of (a) 1/sqrt(36-x^2), (b) x/(36+x^2), and (c) 1/sqrt(x^2-36) using a trig substitution.
Lecture 14
Trig Substitution 2 - Integral for (1+x^2)^{5/2}
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Trig Substitution 2 - Integral for (1+x^2)^{5/2}
Worked problem in calculus. Using a trig substitution, find the indefinite integral int dx/(1+x^2)^{5/2}.
Lecture 15
Trig Substitution 3 - Integral of x^2/sqrt(1-4x^2)
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Trig Substitution 3 - Integral of x^2/sqrt(1-4x^2)
Worked problem in calculus. The trig substitution x = (1/2)sin(theta) is used to find the indefinite integral of f(x) = x^2/sqrt(1-4x^2).
Lecture 16
Trig Substitution 4 - Integral of sqrt(e^{2x} - 1)
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Trig Substitution 4 - Integral of sqrt(e^{2x} - 1)
Worked problem in calculus. Calculate the indefinite integral int sqrt(e^{2x} - 1) dx using a trig substitution for secant.
Lecture 17
Integration with Partial Fractions 1 - Distinct Linear Factors
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Integration with Partial Fractions 1 - Distinct Linear Factors
Worked problem in calculus. Partial fraction integral with distinct linear factors. (a) int dx/(x^2-4), (b) int dx/(x^3-x).
Lecture 18
Integration with Partial Fractions 2 - Repeated Linear Factors
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Integration with Partial Fractions 2 - Repeated Linear Factors
Worked problem in calculus. Calculate the indefinite integral using partial fractions with repeating linear factors for f(x) = 1/(4x^3-4x^2+x).
Lecture 19
Integration with Partial Fractions 3 - Distinct Mixed Factors
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Integration with Partial Fractions 3 - Distinct Mixed Factors
Worked problem in calculus. The indefinite integral of f(x) = (3x^2+2)/(x^2+1)(x^2-x) is computed using partial fractions with distinct linear/quadratic factors. Two methods are presented for obtaining constant terms.
Lecture 20
Integration with Partial Fractions 4 - Repeated Quadratic Factors
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Integration with Partial Fractions 4 - Repeated Quadratic Factors
Worked problem in calculus. Calculate the indefinite integral of f(x) = (x^3-x)/(x^2+1)^2 using partial fractions with repeated quadratic factors.
Lecture 21
Integration with Partial Fractions 5 - Composition with e^x
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Integration with Partial Fractions 5 - Composition with e^x
Calculus: Use partial fractions to find the indefinite integral of f(x) = (e^{3x} + 2e^{2x} + 4e^x)/[(e^{2x}+1)(e^x+1)].
Lecture 22
L'Hopital's Rule 1 - Rational Functions
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L'Hopital's Rule 1 - Rational Functions
Worked problem in calculus. L'Hopital's Rule is presented and examples with algebraic functions are computed. (a) f/g = (x^2 + x)/(x^3-x) ay x=-1, (b) f/g = (2x^2 + x+1)/(4x^2-6) at x= infty, and (c) f/g = x/sqrt(1+x^2) at x=infty.
Lecture 23
L'Hopital's Rule 2 - Trig Limits
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L'Hopital's Rule 2 - Trig Limits
Worked problem in calculus. L'Hopital's rule is applied to the trig limits (a) sin(x)/x at x=0, (b) (cos(x) - 1)/x at x=0, (c) (cos(x) - 1)/x^2 at x=0, and (d) sin(3x)/sin(5x) at x=0.
Lecture 24
L'Hopital's Rule 3 - exp, log, and inverse sine
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L'Hopital's Rule 3 - exp, log, and inverse sine
Worked problem in calculus. L'Hopital's Rule is applied to (a) ln(x)/x at x=infty, (b) (1+x-e^x)/x^2 at x= 0, and (c) (sin^{-1}(x) - pi/2)/(x-1) at x=1-.
Lecture 25
L'Hopital's Rule 4 - Special Indeterminate Forms
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L'Hopital's Rule 4 - Special Indeterminate Forms
Worked problem in calculus. Special indeterminate forms are considered for L'Hopital's Rule. We consider the limits: (a) xln(x) at x=0 (0.infty), (b) (1-x)^{1/sin(x)} at x=0 (1^infty), (c) x^{x^2+x} (0^0) and (d) ln(1-x)/x^3 + (2+x)/2x^2 at x=0 (infty - infty).
Lecture 26
Growth of Functions at Infinity
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Growth of Functions at Infinity
Calculus: We use L'Hopital's Rule to compare the growth at infinity of x^{1/n} and ln(x).
Lecture 27
Improper Integrals 1 - Infinite Limits of Integration
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Improper Integrals 1 - Infinite Limits of Integration
Calculus: We define improper integrals with infinite limits of integration. Examples presented are (a) int_0^infty e^{-x} dx, (b) int_1^infty dx/x, (c) int_1^infty ln(x)/x^2 dx, and (d) int_{-infty}^{infty} dx/(1+x^2).
Lecture 28
Improper Integrals 2 - Vertical Asymptote in Interval
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Improper Integrals 2 - Vertical Asymptote in Interval
Calculus: Improper integrals are defined when a vertical asymptote falls into the range of integration. Examples considered are (a) int_0^1 dx/sqrt(1-x^2), (b) int_0^1 dx/sqrt(x), and (c) int_{-1}^1 dx/x^2.
Lecture 29
Improper Integral of 1/x^p
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Improper Integral of 1/x^p
Calculus: We consider the improper integral of x^{-p} over the region from x=1 to x=infty. Two cases emerge, with divergence of the integral when p is less than 1.
Lecture 30
Mean of the Exponential Distribution
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Mean of the Exponential Distribution
Calculus: We consider an application of improper integrals in probability theory. We find the mean for the probability density function rho(x) = e^{-x} over the region x=0 to x= infty. (Review moments and center of mass of a rod of nonuniform density - Moments 5)