Math 3B: Lecture 11. Noah White. October 25, 2017

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1 Math 3B: Lecture 11 Noah White October 25, 2017

2 Introduction Midterm 1

3 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good.

4 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Grades are online.

5 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Grades are online. TAs have your exams

6 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Grades are online. TAs have your exams Homework

7 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Grades are online. TAs have your exams Homework Two problems to hand in Friday 3 November

8 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Grades are online. TAs have your exams Homework Two problems to hand in Friday 3 November Problem 7, problem set 4

9 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Grades are online. TAs have your exams Homework Two problems to hand in Friday 3 November Problem 7, problem set 4 Problem 3, problem set 5

10 Integration by substitution As we have seen, some antiderivatives are difficult to guess,

11 Integration by substitution As we have seen, some antiderivatives are difficult to guess, especially if it involves reversing the chain rule.

12 Integration by substitution As we have seen, some antiderivatives are difficult to guess, especially if it involves reversing the chain rule. We solve this by introducing a new variable.

13 Integration by substitution As we have seen, some antiderivatives are difficult to guess, especially if it involves reversing the chain rule. We solve this by introducing a new variable. Substitution Suppose u = g(x), then f (g(x))g (x) dx = f (u) du

14 Integration by substitution As we have seen, some antiderivatives are difficult to guess, especially if it involves reversing the chain rule. We solve this by introducing a new variable. Substitution Suppose u = g(x), then f (g(x)) du dx dx = f (g(x))g (x) dx = f (u) du

15 Example 1 Question 4x x dx

16 Example 1 Question 4x x dx Solution We use the substitution u = x 2 + 1, so du dx = 2x, we can write the integral 2 x u x dx = 2 du

17 Example 1 Question 4x x dx Solution We use the substitution u = x 2 + 1, so du dx = 2x, we can write the integral 2 x u x dx = 2 du = u C

18 Example 1 Question 4x x dx Solution We use the substitution u = x 2 + 1, so du dx = 2x, we can write the integral 2 x u x dx = 2 du = u C = 4 3 (x 2 + 1) C

19 Integration by substitution (definite integrals) Substitution for definite integrals Suppose u = g(x), then b a f (g(x))g (x) dx = g(b) g(a) f (u) du

20 Integration by substitution (definite integrals) Substitution for definite integrals Suppose u = g(x), then b a f (g(x))g (x) dx = g(b) g(a) f (u) du Example 1 0 4x x dx = u du

21 Integration by substitution (definite integrals) Substitution for definite integrals Suppose u = g(x), then b a f (g(x))g (x) dx = g(b) g(a) f (u) du Example 1 0 4x x dx = 2 = [ 2 3 u 3 2 u du ] 2 1

22 Integration by substitution (definite integrals) Substitution for definite integrals Suppose u = g(x), then b a f (g(x))g (x) dx = g(b) g(a) f (u) du Example 1 0 4x x dx = u du [ ] 2 2 = 2 3 u ( ) 2 = = 4 3 (2 2 1)

23 The product rule Just like integration by substitution reverses the chain rule, integration by parts "reverses" the product rule: d dx f (x)g(x) = f (x)g(x) + f (x)g (x)

24 The product rule Just like integration by substitution reverses the chain rule, integration by parts "reverses" the product rule: d dx f (x)g(x) = f (x)g(x) + f (x)g (x) writen another way (uv) = u v + uv

25 Integration by parts (uv) = u v + uv

26 Integration by parts (uv) = u v + uv Lets integrate both sides (uv) dx = u v dx + uv dx

27 Integration by parts (uv) = u v + uv Lets integrate both sides (uv) dx = u v dx + uv dx By the fundamental theorem of calculus uv = u v dx + uv dx

28 Integration by parts (uv) = u v + uv Lets integrate both sides (uv) dx = u v dx + uv dx By the fundamental theorem of calculus uv = u v dx + uv dx Rearranging...

29 Integration by parts The integration by parts formula uv dx = uv u v dx

30 Integration by parts The integration by parts formula uv dx = uv u v dx Alternative statement u dv = uv v du

31 Examples One the board...

Math 111 lecture for Friday, Week 10

Math 111 lecture for Friday, Week 10 Math lecture for Friday, Week Finding antiderivatives mean reversing the operation of taking derivatives. Today we ll consider reversing the chain rule and the product rule. Substitution technique. Recall