Homework 3 Techniques of Integration 1. For each of the indefinite integrals below, decide whether the integral can be evaluated using u-substitution (write U-SUB), integration by parts (write BY PARTS), a combination of the two (write COMBO), or neither (write NEITHER). For integrals for which your answer is affirmative, state the substitution(s) you would use. It is not necessary to evaluate the integrals. R Function: ln(x2 + 1) dx R Function: x5 sin(x3 ) dx Technique: Technique: Substitution(s): Substitution(s): Function: R √1 x 1−x2 dx Function: Technique: Technique: Substitution(s): Substitution(s): Function: R √ x 1−x2 ln(x) x dx R Function: x ln(x) dx dx Technique: Technique: Substitution(s): Substitution(s): R Function: x2 sin(x3 ) dx Math 252 R Function: Technique: Technique: Substitution(s): Substitution(s): R 2x+3 1+x2 dx CRN 36208 2. The following are examples of integrals that have converted to a simpler form using substitution or integration by parts, along with the substitution used (in the case of substitution), or one of the two substitutions used (in the case of integration by parts). Write an indefinite integral that is equal to the given one, by working through the substitutions backward. (it may help to check your conjecture by actually calculating the integral you find, although you only need to write down your integral for credit.) (a) Z sin(u) du, u = ln(x) Original Integral: (b) 2 2 x(x + 1)3/2 − 3 3 Z (x + 1)3/2 dx, u=x Original Integral: (c) 1 3 Z 1 du, u u = x3 + 3x Original Integral: (d) − w2 1 cos(10w) + 10 5 Z w cos(10w) dw, dv = sin(10w) dw Original Integral: Math 252 CRN 36208 3. As you saw in the last homework assignment, streaming video has made lectures from major universities readily available and free. But this trend isn’t limited to large universities; Patrick Jones of Austin Community College created PatrickJMT, at http://patrickjmt.com/ by himself. Go to this website and scroll down to the Calculus section. All the topics we will cover this quarter (and more) have video lectures for you to reference. One topic that we will only talk about briefly in class in Week 4 is Integration by Partial Fractions. Find the video Integration by Partial Fractions: Determining Coefficients on PatrickJMT and watch the the first 10 minutes (just the first example). There is also a BRIEF explanation on pages 391-392 in your textbook. (a) Calculate the following definite integral using the video’s first technique of finding the partial fraction numerators A and B. (show ALL work and SIMPLIFY your solution) Z 10 1 dx 2 x −4 4 Solution: (continued on the next page) Math 252 CRN 36208 (b) Calculate the following definite integral using the video’s second technique of finding the partial fraction numerators A and B. (show ALL work and SIMPLIFY your solution)) Z 0 1 2−x dx x2 + 5x Solution: You’re now done with Homework 3. The following is a list of problems that would be good to study for your first midterm at the end of April. Do not turn them in, but please ask questions about any that are troubling. • Section 5.5, #7-36,62,63,64,66,68,72(a) • Section 5.6, #3-24,25-30,39,40 Math 252 CRN 36208