Integration by parts, sometimes multiple times in the same problem. a. Show your choices. b. Evaluate the integral. a.

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1 Instructions Complete the homework with a pencil. Show your work. Use extra paper if necessary. Homework can be turned in as follows Turn in homework in person at KPC Scan and to UAACalculus@yahoo.com (preferred) Regular mail: Marion Yapuncich, Math Department, Kenai Peninsula College, 56 College Road, Soldotna, AK Fax: (Cover sheet appreciated) Integration by parts, sometimes multiple times in the same problem.. xe x dx a. Show your choices d b. Evaluate the integral. xln xdx a. Show your choices d b. Evaluate the integral Page of

2 3. e t sin ω t dt a. Show your choices d b. Evaluate so far c. Integration by parts again. Show your second set of choices: d d. Evaluate the integral, combine like terms and find the answer. Page of

3 3 4. Evaluate x sin xdx by using the reduction formulas u u n n sin aud cosaudu = n u n n cosau + u cosau du a a n u n n sin au u sin au du a a Page 3 of

4 5. xsec xdx a. Show your choices d b. Evaluate the integral c. Show another set of choices d d. Explain why theses were not good choices by showing the problem with evaluating the integral. 6. Demonstration that two methods give the same answer: sin θ cos θ dθ a. Solve, without using integration by parts. b. Integration by parts. Show your choices d c. Evaluate the integral. Set up using integration by parts, but then solve the integral directly. This purpose behind this problem is to show that integration by parts can be applied, even when it is not necessary. Page 4 of

5 3π / 7. a. Rewrite sin( π / ln x ) dx by using change of variables u= ln x to obtain 0.45 e u.55 sinu du. Show steps for any credit. b. (Extra credit) Solve 7a using answer from problem 3 with ω= Page 5 of

6 8. Use partial Fractions, preceded by substitution to solve sec x dx 3 tan x tan x sec x A B C a. Show that, by substitution, dx = du x x tan tan u u u b. Find the solution. In this problem, A = B = and C =. Since this is an indefinite integral, it is necessary to reverse the substitution in the final answer. Page 6 of

7 9. Integrate the following using the multiplication formulas for sine. sin( 6t )sin(0t ) dt. (I understand that there is an error in the youtube lectures for the product indentities for trig functions. I can't find the error. Here are the correct identities: sin Asin B= [cos( A B) cos( A+B)] sin A cos B= [sin( A+B)+sin ( A B)] 0. Integrate the following using the multiplication formulas for sine and cosine: sin(t )cos(5t ) dt Page 7 of

8 . Using the trigonometric substitution x = 5sec θ, a. Solve dx x x 5 b. Use the appropriate triangle (see Lecture Notes), to reverse the substitution in the answer. Page 8 of

9 . Show that the following improper integral diverges x dx x Show that the following improper integral has the value of. (Vertical Asymptote at x =.) 0 dx x + 4. The integral dx x p diverges for p and converges to p > Hint: Steps are shown in the text and in the Lecture Notes for Improper Integrals. a. Demonstrate this result by calculating the integral for p = / b. Demonstrate the converges by calculating that integral for p =. p for Page 9 of

10 5. Numerical Integration Calculate the integral 3 e t dt by hand (not by computer program) using the Midpoint and Trapezoid s Rule with a mesh size of N = 4. a. Calculate the step size, h. b. Write the Midpoint rule. c. Use the Midpoint rule to approximate the integral. e. Use the Trapezoidal rule to approximate the integral. d. Write Simpson s Rule Page 0 of

11 6. (Compare step sizes for each method) Use the expressions for the error of the Midpoint, Trapezoidal and Simpson s rule to find the following results for maximum mesh size h and number of meshes N 3 for a desired accuracy of E d = 0 for each method for the integral 3 3 e t dt. The equations for finding h and N for each method from the desired accuracy are give at the end of the written lecture 7F Numerical Integration. Hint: You will need the maximum value on [,3] of the second derivative for the Midpoint and Trapezoidal Rules and the fourth derivative for Simpson s Rule. Work must be shown. Midpoint h = N = 4930 Trapezoid h = N = 6973 Simpson's h = 0.09 N = 04 Page of