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- 1. Merit Worksheet 22. Math 221 ED1. Fall 2015. 11/13/2015. 5.5 Substitution Rule 1. Read: The substitution rule is the most fundamental way of solving integrals, so we have to practice a lot. The substitution rule is basically the opposite of the chain rule. Therefore it is a lot harder and therefore a lot of integrals cannot be solved 2. Write down the indefinite versions of the substitution formula: f(g(x))g (x) dx = dx f(u) du = dx where u is a function in terms of x. 3. Enhance your answer by finding the definite versions of the substitution formula: b a f(g(x))g (x) dx = dx b a f(u) du = dx where u is a function in terms of x. Now some actual questions, so make sure everybody is at the same spot when you start them (yayyyyyyyyyyyyyyyyyy): 4. Solve this integral in two ways: (a) Find the following integral using standard techniques from last week: 3 1 (x − 1) dx = (b) (Board Questions) Draw the area and the shape of the integral you just computed . (c) Use the substitution u = x − 1 to convert this integral to another: 3 1 (x − 1) dx = du (d) (Board Questions) Draw the area and the shape of the integral you just got from substi- tution. (e) How do the two shapes compare to each other? What conclusion can you draw from this? What does it tell you about substitution? 5. Let f(x) = x2 − 4x + 4. Go through the same steps as the last question to find the integral of f(x) from 2 to 4. Use u = x − 2 for part (c). Now we will finally start doing some questions where it helps to solve them with substitution. Make sure everybody in the group is here together. More yayyyyyyyyyy.
- 2. 6. Solve the following indefinite integral using the substitution u = x2 : 2x sin(x2 ) dx 7. Solve the following definite integral using the substitution u = x3 : 2 1 x2 ex3 dx 8. Sometimes there is not necessarily one correct substitution. Find 1 √ r(1 + √ r)2 dr in two ways by using these two substitutions: u = √ r and u = 1 + √ r. From now on you have to find u yourselves. Make sure everybody made it till here!!! 9. Find 1 1 + 1 e 1 √ e de 10. Find sin3 (t) + 4 sin7 (t) dt 11. Find sin5 (x) cos(x) dx 12. Find π 4 0 tan6 (x) sec2 (x) dx 13. This one is harder. Find 1 x(ln(x))5 dx 14. Can you do this one? Find 1 0 t(1 − t)100 dt 15. This one is tricky. Find π 2 0 sin3 (p) dp 16. Let’s go meta. Let f(x) be a differentiable function such that f(2) = 0 and f(5) = e8 . Find 5 2 f (t) f(t) dt 17. This one is so hard I will again tell you what u to use. Use u = sin(t) to find the following integral: 1 √ 1 − t2 dt 18. This one’s similar. Use u = tan(t) to find the following integral: 1 1 + t2 dt 19. Now some really hard ones. Find 1 t2 − 2t + 2 dt 20. Final one: Find 1 √ 4t − 4t2 dt