  # Calculus with Dr. Bob V: Advanced Integration Techniques

## Video Lectures

Displaying all 30 video lectures.
 Lecture 1 Play Video Int by Parts 1 - Natural Log and ExponentialWorked problem in calculus. Calculate the indefinite integrals of: (a) x^3 ln(x) and (b) x^2 e^{-x}. Lecture 2 Play Video Int by Parts 2 - Trig FunctionsWorked problem in calculus. Find the indefinite integrals of the trig functions: (a) tan^{-1}(x) and (b) xsec^2(x). Lecture 3 Play Video Int by Parts 3 - Definite IntegralsWorked 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 Play Video 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 Play Video 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 Play Video 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 Play Video 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 Play Video 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 Play Video 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 Play Video 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 Play Video 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 Play Video 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 Play Video Trig Substitution 1 - Basic Inverse Trig IntegralsWorked 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 Play Video 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 Play Video 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 Play Video 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 Play Video Integration with Partial Fractions 1 - Distinct Linear FactorsWorked problem in calculus. Partial fraction integral with distinct linear factors. (a) int dx/(x^2-4), (b) int dx/(x^3-x). Lecture 18 Play Video Integration with Partial Fractions 2 - Repeated Linear FactorsWorked 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 Play Video Integration with Partial Fractions 3 - Distinct Mixed FactorsWorked 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 Play Video Integration with Partial Fractions 4 - Repeated Quadratic FactorsWorked 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 Play Video Integration with Partial Fractions 5 - Composition with e^xCalculus: 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 Play Video L'Hopital's Rule 1 - Rational FunctionsWorked 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 Play Video L'Hopital's Rule 2 - Trig LimitsWorked 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 Play Video L'Hopital's Rule 3 - exp, log, and inverse sineWorked 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 Play Video L'Hopital's Rule 4 - Special Indeterminate FormsWorked 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 Play Video Growth of Functions at InfinityCalculus: We use L'Hopital's Rule to compare the growth at infinity of x^{1/n} and ln(x). Lecture 27 Play Video Improper Integrals 1 - Infinite Limits of IntegrationCalculus: 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 Play Video Improper Integrals 2 - Vertical Asymptote in IntervalCalculus: 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 Play Video Improper Integral of 1/x^pCalculus: 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 Play Video Mean of the Exponential DistributionCalculus: 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)