Worksheet Week 1 Review of Chapter 5, from Definition of integral to Substitution method

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1 Worksheet Week Review of Chapter 5, from Definition of integral to Substitution method This worksheet is for improvement of your mathematical writing skill. Writing using correct mathematical expressions and steps is really important part of doing math. Please print out this worksheet and try to solve problems, following given steps. You don t need to submit this worksheet. It is not a homework. This worksheet don t guarantee a high score on exam. You have to work with many problems in the textbook. This is just a guideline for writing of answer. I will upload a model solution of the worksheet. Please compare it with your own answer. It is possible that you can have a different solution to the model solution. Don t worry about it - your solution will be accepted too, if there is no mathematical error. But compare the model solution with yours. If you have any questions, do not hesitate contact me.. Evaluate the definite integral 4x x 2. (a) Simplify the integrand if you can (for instance, use law of exponents). 4x x 2 = 4 + 3x 2 Sometimes such algebraic manipulation is very important to computation. (b) Find the antiderivative and use the fundamental theorem of calculus x 2 = [ 4x 3x ] 3 (c) Compute the answer. [ 4x 3x ] 3 = (4 3 3 ) (4 3 ) = 3

2 2. Evaluate the definite integral = [ ] 3 x3 + 2x x 5 2x x 3 = x 5 2x x 3. x 2 2x 2 = ( 3 2) ( 3 ( 27) + 2 ) = Evaluate the definite integral π ( + cos x). π ( + cos x) = [x + sin x] π = (π + sin π) ( + sin ) = π 2

3 4. (Substitution method for indefinite integral) Evaluate the indefinite integral x(x 2 + 3) 2. (a) Define an appropriate variable u in terms of x. u = x (b) Compute in terms of x. Note that the symbol is. = 2x = 2x (c) Substitute everything on the integral using u and. x(x 2 + 3) 2 = (x 2 + 3) 2 x = u 2 2 After this step, there must be no x in your integral. (d) Compute given indefinite integral. u 2 u3 = C (e) Replace u by a function of x you ve chosen in (a). u C = (x2 + 3) 3 + C 6 Unfortunately, there is no golden rule for finding u, but here is a tip for you: In step (c), you have to change everything in terms of u. So you must find u such that this substitution is possible. In particular, ( ) 2, ( or its constant multiple (not ) 3, ) must be in the integrand. In other words, find u such that you can express given integrand as something about u times. One another suggestion is that define u as the inside of complicate part, for instance inside of parentheses, an exponential function or a square root. 3

4 5. Evaluate the indefinite integral, following similar steps in previous problem. x 5 (x 6 2) 25 u = x 6 2 = 6x5 = 6x 5 x 5 (x 6 2) 25 = (x 6 2) 25 x 5 = u 25 6 = u C = u C = (x6 2) 26 + C Evaluate the indefinite integral, following similar steps in previous problem. x 2 e x3 u = x 3 = 3x2 = 3x 2 x 2 e x3 = e x3 x 2 = e u 3 = 3 eu + C = 3 ex3 + C 4

5 7. (Substitution method for definite integral) Evaluate the definite integral e 7 x( + ln x). (a) Define appropriate variable u in terms of x. u = + ln x (b) Compute in terms of x. = x = x (c) Find two endpoints u() and u(e 7 ). u() = + ln =, u(e 7 ) = + ln e 7 = + 7 = 8 (d) Substitute everything (including endpoints) on the integral using u and. e 7 e 7 x( + ln x) = 8 ( + ln x)x = u Again, when you apply the substitution method, you must replace everything about x by something about u. (e) Compute the definite integral. 8 u = [ln u ]8 = ln 8 ln = ln 8 5

6 8. Evaluate the definite integral 2x (4 + x 2 ) 2 following similar steps in the previous problem. u = 4 + x 2 = 2x = 2x u() = 4, u() = 4 + = 5 2x 5 (4 + x 2 ) 2 = 4 u 2 = 5 4 u 2 = [ u ] 5 4 = 5 ( 4 ) = = 2 9. Evaluate the definite integral x x 2 following similar steps in the previous problem. u = x 2 = 2x = 2x, = x 2 u() =, u() = = x x 2 = = u u( 2 ) = 2 = u 2 2 [ ] 3 u 3 2 = 3 = 3 6