1 -Introduction to the course (138.18 MB)
2 -Two types of integrals, two ways to go (110.38 MB)
3 -My choices versus the choices in the book (142.56 MB)
4 -The spoiler you need to follow both paths (217.81 MB)
5 -Main integration techniques and where to find them (94.11 MB)
6 -Plenty of applications (52.52 MB)
1 -Finding values of some integrals with help of geometry (128.42 MB)
2 -Integrals of odd functions over compact and symmetric-to-zero intervals (55.34 MB)
3 -Integrals of even functions over compact and symmetric-to-zero intervals (76.74 MB)
4 -Integrals of periodic functions (46.34 MB)
5 -Some nice properties of the mean value (281.8 MB)
6 -Optional What about integrals equal to zero (121.78 MB)
1 -The connection between finding areas and finding antiderivatives (104.43 MB)
10 -Computing average values of functions on intervals, an example (317.98 MB)
11 -Integration by parts for Riemann integrals, two ways to go (183.99 MB)
12 -Integration by substitution for Riemann integrals, two ways to go (346.25 MB)
13 -An illustration for integration by substitution (113.53 MB)
14 -Applications of properties of integrals, an exercise (405.34 MB)
15 -Limits of some type of sequences, Example 1 (279.46 MB)
16 -Limits of some type of sequences, Example 2 (201.4 MB)
17 -Limits of some type of sequences, Example 3 (140.04 MB)
18 -Limits of some type of sequences, Example 4 (135.33 MB)
19 -Limits of some type of sequences, Example 5 (239.93 MB)
2 -Algebraic and transcendental functions (53.74 MB)
20 -Differentiating functions defined with help of integrals, Example 1 (102.98 MB)
21 -Differentiating functions defined with help of integrals, Example 2 (334.91 MB)
22 -Future Fundamental Theorems in Multivariable Calculus (61.36 MB)
3 -What came first the logarithm or the exponential (157.86 MB)
4 -Function of the upper limit of integration its continuity (146.36 MB)
5 -Fundamental Theorem of Calculus, part 1 (136.67 MB)
6 -Fundamental Theorem of Calculus, part 2 (the evaluation theorem) (96.96 MB)
7 -Area of a disc; another method for computing integrals from V147, V151, and V155 (451.44 MB)
8 -Evaluating integrals, another example (294.55 MB)
9 -Important integrals for future applications (Fourier series) (338.75 MB)
1 -Area between the graph of a function and the x-axis (99.05 MB)
2 -Another (than in V36, V108, V114) method for finding antiderivative of secant (408.09 MB)
3 -Area between two graphs (247.46 MB)
1 -Arc length derivation of the formula, some examples (277.48 MB)
2 -Arc length, Problem 1 (459.26 MB)
3 -Arc length, Problem 2 (195.16 MB)
1 -Rotational volume, different situations and different methods (97.39 MB)
2 -The disk method derivation and an example (74.74 MB)
3 -We are finally able to confirm two well-known formulas for volume (226.69 MB)
4 -The washer method derivation and an example (77.21 MB)
5 -Cylindrical shells derivation and an example (111.77 MB)
6 -Cylindrical shells for a domain between two graphs (53.47 MB)
1 -Rotational surface area derivation and two examples (291.17 MB)
2 -Rotational surface area, Problem 1 (357.49 MB)
3 -Rotational surface area, Problem 2 (266.59 MB)
1 -Improper integrals, an introduction (211.8 MB)
2 -Improper integrals of the first kind, Problem 1 (177.59 MB)
3 -Improper integrals of the first kind, Problem 2 (282.28 MB)
1 -Improper integrals of the second kind (213.4 MB)
2 -Improper integrals of the second kind, Problem 1 (179.95 MB)
3 -Improper integrals of the second kind, Problem 2 (351.05 MB)
1 -Comparison criteria for improper integrals of non-negative functions (494.3 MB)
2 -Comparison criteria, Problem 1 (182.87 MB)
3 -Comparison criteria, Problem 2 (376.36 MB)
4 -Wrap-up and some words about Calculus 2, part 2 Sequences and series (34.94 MB)
1 -Bonus Lecture (40.71 MB)
1 -Reverting differentiation in simple cases (75.06 MB)
10 -Linearity of integration, Exercise 6 (154.03 MB)
11 -Linearity of integration, Exercise 7 (51.31 MB)
12 -Linearity of integration, Exercise 8 (139.53 MB)
13 -Linearity of integration, Exercise 9 (150.64 MB)
14 -Linearity of integration, Exercise 10 (101.11 MB)
15 -Linearity of integration, Exercise 11 (102.81 MB)
16 -Linearity of integration, Exercise 12 (212.9 MB)
17 -Linearity of integration, Exercise 13 (86.69 MB)
18 -Linearity of integration, Exercise 14 (155.77 MB)
19 -Linearity of integration, Exercise 15 (71.92 MB)
2 -Some important facts about primitive functions (51.09 MB)
20 -A soft introduction to variable substitution (95.47 MB)
21 -Easy variable substitution, Exercise 16 (155.69 MB)
22 -Easy variable substitution, Exercise 17 (144.52 MB)
23 -Easy variable substitution, Exercise 18 (135.99 MB)
24 -Easy variable substitution and some uneasy trigonometry, Problem 1 (97.8 MB)
25 -Easy variable substitution and some uneasy trigonometry, Problem 2 (210.03 MB)
26 -Easy variable substitution, Exercise 19 (217.86 MB)
27 -Easy variable substitution, Exercise 20 (118.96 MB)
28 -Logarithmic derivative and its charm (22.9 MB)
29 -Logarithmic derivative, Exercise 21 (89.5 MB)
3 -Integrals of hyperbolic functions and some related stuff (50.73 MB)
30 -Logarithmic derivative three difficult and important examples, Problem 3 (278.17 MB)
31 -The last one, Exercise 22 (97.62 MB)
4 -Linearity of integration (54.59 MB)
5 -Linearity of integration, Exercise 1 (152.59 MB)
6 -Linearity of integration, Exercise 2 (168.11 MB)
7 -Linearity of integration, Exercise 3 (185.82 MB)
8 -Linearity of integration, Exercise 4 (170.52 MB)
9 -Linearity of integration, Exercise 5 (147.58 MB)
1 -Integration by parts how it works and when to use it (77.92 MB)
10 -Integration by parts, Exercise 1 (183.36 MB)
11 -Integration by parts, Exercise 2 (239.56 MB)
12 -Integration by parts, Exercise 3 (107.51 MB)
13 -Integration by parts, Exercise 4 (216.72 MB)
14 -Integration by parts, Exercise 5 (116.65 MB)
15 -Integration by parts, Exercise 6 (495.38 MB)
2 -Integration by parts Example 1 (148.98 MB)
3 -Integration by parts Example 2 (109.17 MB)
4 -Integration by parts Example 3 (82.07 MB)
5 -Integration by parts Example 4 (287.77 MB)
6 -Integration by parts Example 5 (130.51 MB)
7 -Integration by parts Example 6 (106.88 MB)
8 -Integration by parts Example 7 (151.6 MB)
9 -What happens when the degree of the polynomial is higher (48.02 MB)
1 -Integration by substitution how it works and when to use it, Example 1 (155.5 MB)
10 -Recognising (almost) derivatives, Example 10 (54.78 MB)
11 -Recognising (almost) derivatives, Example 11 (111.87 MB)
12 -Recognising (almost) derivatives, Example 12 (60.85 MB)
13 -Recognising (almost) derivatives, Example 13 (111.66 MB)
14 -Back to the integral from V49 for n=-1, Example 14 (60.24 MB)
15 -Back to the integral from V30, Problem 1 (208.3 MB)
16 -Different results can happen how to handle them, Problem 2 (64.98 MB)
17 -Optional Back to the integral of cosecant from V36, Problem 3 (343.34 MB)
18 -A less obvious case, Problem 4 (80.53 MB)
19 -A less obvious case, Problem 5 (128.92 MB)
2 -Easy substitutions from Section 2, Example 2 (325.11 MB)
20 -Three examples related to the arctangent, Problem 6 (247.48 MB)
21 -Back to arcsine, Problem 7 (212.92 MB)
22 -A strange one, Problem 8 (204.51 MB)
23 -Three examples with the square root of x (150.38 MB)
3 -Recognising (almost) derivatives, Example 3 (164.19 MB)
4 -Recognising (almost) derivatives, Example 4 (75.4 MB)
5 -Recognising (almost) derivatives, Example 5 (87.45 MB)
6 -Recognising (almost) derivatives, Example 6 (42.74 MB)
7 -Recognising (almost) derivatives, Example 7 (86.86 MB)
8 -Recognising (almost) derivatives, Example 8 (98.4 MB)
9 -Recognising (almost) derivatives, Example 9 (59.94 MB)
1 -Five key concepts needed for the integration of rational functions (51.7 MB)
10 -Variable substitution, logarithm, and arctangent, Formula 6 (372.5 MB)
11 -Method of Strategic Substitution in partial fraction decomposition (350.35 MB)
12 -Integration of rational functions, Exercise 1 (389.75 MB)
13 -Integration of rational functions, Exercise 2 (150.86 MB)
14 -Integration of rational functions, Exercise 3 (387.66 MB)
15 -Integration of rational functions, Exercise 4 (288.57 MB)
16 -Integration of rational functions, Exercise 5 (256.03 MB)
17 -Integration of rational functions, Problem 1 (428.09 MB)
18 -Integration of rational functions, Problem 2 (581.34 MB)
19 -Integration of rational functions, Problem 3 (154.77 MB)
2 -Prerequisites from Precalculus 2 (287.92 MB)
20 -Integration of rational functions, Problem 4 (180.43 MB)
3 -We have already worked with integrals of rational functions (138.01 MB)
4 -Integrals leading to the logarithm or to power functions, Formula 1 (193.72 MB)
5 -Integrals leading to the logarithm or to power functions, Formula 2 (282.97 MB)
6 -Integrals leading to the arctangent or to power functions, Formula 3 (273.17 MB)
7 -Formula 3, Example 3 (158.98 MB)
8 -Variable substitution and arctangent, Formula 4 (54.83 MB)
9 -Variable substitution and arctangent, Formula 5 (45.95 MB)
1 -Trigonometric formulas and where to find them (49 MB)
10 -The product of powers of the sine and the cosine (60.37 MB)
11 -The product of powers of the sine and the cosine, Exercise 3 (75.73 MB)
12 -The product of powers of the sine and the cosine, Exercise 4 (98.27 MB)
13 -Integrals involving secants and tangents, Exercise 5 (337.5 MB)
14 -Recursive formulas for integrals of powers of the tangent or of the secant (127.93 MB)
15 -Product to sum formulas are good for integrals (40.76 MB)
16 -Rational expressions in two variables (132.23 MB)
17 -World's sneakiest substitution the universal substitution (312.91 MB)
18 -The universal substitution, Exercise 6 (93 MB)
19 -The universal substitution, Exercise 7 (213.37 MB)
2 -Trigonometric integrals we have seen until now (102.34 MB)
20 -The universal substitution, Exercise 8 (152.14 MB)
21 -The universal substitution, Exercise 9 (321.41 MB)
22 -Rational expressions odd w r t the sine, Exercise 10 (312.64 MB)
23 -Rational expressions odd w r t the sine, Exercise 11 (248.65 MB)
24 -Rational expressions odd w r t the sine, Exercise 12 (186.73 MB)
25 -Rational expressions odd w r t the cosine, Exercise 13 (256.95 MB)
26 -Rational expressions even w r t both variables, Exercise 14 (693.73 MB)
27 -Rational expressions even w r t both variables, Exercise 15 (403.02 MB)
28 -The last one, just for fun, Problem 1 (129.97 MB)
3 -Power reduction formulas for computing integrals of the sine (or cosine) squared (29.39 MB)
4 -Integral of the cube of the sine or of the cosine, by change of variables (250.9 MB)
5 -Integral of the fourth power of the sine with help of repeated power reduction (75.78 MB)
6 -Recursive formulas for integrals of powers of the sine or of the cosine (360.56 MB)
7 -Powers of the sine or of the cosine versus the functions of multiple arguments (82.18 MB)
8 -Cubes in two ways, Exercise 1 (131.16 MB)
9 -Fourth powers in two ways, Exercise 2 (202.25 MB)
1 -Nobody will teach you all the integration techniques, but (76.21 MB)
10 -Optional A geometrical interpretation of Euler's substitutions 2 and 3 (106.26 MB)
11 -Euler substitutions, Problem 1 (338.37 MB)
12 -Euler substitutions, Problem 2 (360.17 MB)
13 -Euler substitutions, Problem 3 (379.62 MB)
14 -Rational expressions of rational powers (50.94 MB)
15 -Rational expressions of rational powers, Problem 4 (354.11 MB)
16 -Rational expressions of rational powers, Problem 5 (85.88 MB)
17 -Rational expressions of rational powers, Problem 6 (80.19 MB)
18 -Rational expressions of rational powers, Problem 7 (206.69 MB)
19 -Rational expressions of rational powers, an atypical one, Problem 8 (271.62 MB)
2 -Euler substitutions, why three cases are enough (91.68 MB)
20 -Direct (u) versus inverse substitution (82.63 MB)
21 -Back to trigonometric substitutions from Section 6 reference triangles (129.67 MB)
22 -Three triangle substitutions (59.91 MB)
23 -Triangle substitutions, Case 1 (84.74 MB)
24 -Triangle substitution 1, an example (223.26 MB)
25 -Triangle substitution 1, Problem 9 (159.05 MB)
26 -Triangle substitution 1, Problem 10 (142.55 MB)
27 -Triangle substitutions, Case 2 (51.95 MB)
28 -Triangle substitution 2, an example (209.04 MB)
29 -Triangle substitution 2, Problem 11 (213.51 MB)
3 -Euler substitutions, why they work (42.91 MB)
30 -Triangle substitution 2, Problem 12 (131.39 MB)
31 -Triangle substitutions, Case 3 (57.73 MB)
32 -Triangle substitution 3, an example (278.1 MB)
33 -Triangle substitution 3, Problem 13 (184.58 MB)
34 -Undetermined coefficients (100.9 MB)
35 -Some remarks about equality of certain functions (176.72 MB)
36 -Undetermined coefficients instead of integration by parts, an example (228.21 MB)
37 -Undetermined coefficients instead of integration by parts, another example (515.5 MB)
38 -Optional Our most complicated method (62.1 MB)
39 -Optional Our most complicated method, an example (482.86 MB)
4 -Euler's substitution 1, an explanation (114.48 MB)
40 -Optional Our most complicated method, Problem 14 (430.85 MB)
5 -Euler's substitution 1, an example (493.98 MB)
6 -Euler's substitution 2, an explanation (74.21 MB)
7 -Euler's substitution 2, an example (365.38 MB)
8 -Euler's substitution 3, an explanation (46.33 MB)
9 -Euler's substitution 3, an example (348.19 MB)
1 -Practice, practice, practice (57.39 MB)
10 -Integrals, Problem 9 (139.49 MB)
11 -Integrals, Problem 10 (215.12 MB)
12 -Integrals, Problem 11 (259 MB)
13 -Integrals, Problem 12 (203.25 MB)
14 -Integrals, Problem 13 (103.61 MB)
15 -Integrals, Problem 14 (129.67 MB)
16 -Integrals, Problem 15 (74.9 MB)
17 -Integrals, Problem 16 (294.55 MB)
18 -Integrals, Problem 17 (360.21 MB)
19 -A very brief introduction to Initial-Value Problems (IVP) (124.49 MB)
2 -Integrals, Problem 1 (198.2 MB)
20 -IVP verifying solutions, an example (154.13 MB)
21 -IVP finding solutions, an example (101.57 MB)
22 -Position, velocity, acceleration (102.37 MB)
23 -Position, velocity, acceleration, Problem 18 (310.94 MB)
24 -IVP Falling under gravity, Problem 19 (283.92 MB)
25 -Direct versus inverse problems (51.15 MB)
3 -Integrals, Problem 2 (257.61 MB)
4 -Integrals, Problem 3 (149.43 MB)
5 -Integrals, Problem 4 (103.37 MB)
6 -Integrals, Problem 5 (120.98 MB)
7 -Integrals, Problem 6 (142.59 MB)
8 -Integrals, Problem 7 (164.09 MB)
9 -Integrals, Problem 8 (175.84 MB)
1 -From Geometry to Calculus, one more time (116.28 MB)
10 -Continuous functions on compact intervals are uniformly continuous (176.17 MB)
11 -Continuous functions on compact intervals are Riemann integrable (61.46 MB)
12 -Monotone functions on compact intervals are Riemann integrable (56.13 MB)
13 -Some properties of oscillations and oscillatory sums (194.25 MB)
14 -Some properties of Riemann integrals (178.12 MB)
15 -Monotonicity of integrals (72.5 MB)
16 -Additivity of integration w r t the interval (179.94 MB)
17 -Integrability of piecewise continuous functions (140.83 MB)
18 -Mean Value Theorem for integrals 1 (117.39 MB)
19 -Mean Value Theorem for integrals 2 (116.43 MB)
2 -The concept of area (126.42 MB)
20 -Mean value of a continuous function over a compact interval (57.08 MB)
3 -Our early example (56.78 MB)
4 -Integrability (170.83 MB)
5 -Refinements of partitions, and relations between upper and lower Riemann sums (235.75 MB)
6 -Integrable functions, an example (64.37 MB)
7 -Finally, an example of a function that is not integrable (59.68 MB)
8 -More practical tests for integrability (Cauchy, sequential) (210.8 MB)
9 -Application of the new test to the example from V191 (67.3 MB)