Section 7.1 Integration by Substitution


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1 Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find n ntiderivtive using ny method. Once you hve figured out which one this is, either use your clcultor to evlute it or pproximte it using n = 4 rectngles sinxcos xdx xsin (x 2 + 5)dx e /x x 2 dx tnxdx x 3 x 2 dx e x e x + dx + 4x 2 dx xln x dx e x2 dx 0. x 2 x dx Generl Rules for Substitution. In most cses, you will wnt to choose w so tht dw is sitting somewhere else in your integrnd. 2. It is often helpful to choose w to the be the inside of some other function. CAUTION: In order for the substitution rule to work properly, you must remember to write down your dx s, dw s, etc., in your integrls.
2 Recll the Product Rule for differentition: d dx [u(x)v(x)] = Section 7.2 Integrtion by Prts Integrtion by Prts Formul. Exmples:. xcosxdx
3 2. xlnxdx Notes on Integrtion by Prts:. u nd v together must give the entire integrnd; tht is, the entire function you re integrting. 2. If possible, we choose u nd v so tht the integrl uv dx cn be evluted directly. Otherwise, we try to choose u nd dv so tht uv dx is simpler thn the integrl we strted with. 3. Generl Rule. Choose v to be the lrgest portion of the originl integrl tht you cn find n ntiderivtive for. This rule won t lwys work, but will help guide you to good choice in mny cses. Exercises Evlute ech of the following integrls.. xe x dx 2. x 2 e 2x dx 3. e lnxdx x 5 e x3 dx rcsinxdx e x sinxdx
4 Section 7.3 Tbles of Integrls Exmple. By competing the squre nd substituting, evlute x 2 + 2x + 6 dx. Exmple 2. Evlute ech of the following, using integrl tbles s pproprite. () 4x2 9 dx (b) x 2 sin xdx
5 Exmple 3. Clculte sin 3 xdx. Exmple 4. Clculte cos 5 xsin 2 xdx. Exmple 5. Clculte sin 2 xdx.
6 Integrting Powers of Sine nd Cosine: cos m xsin n xdx Cse. If m is odd, stel power from cosx nd let u = sin x. Cse 2. If n is odd, stel power from sinx nd let u = cosx. Cse 3. If both m nd n re even, you will need to use the identity sin 2 x = 2 ( cos(2x)) nd/or the identity cos 2 x = 2 ( + cos(2x)). Exmples nd Exercises. Evlute ech of the following, using integrl tbles where pproprite. () cos 3 (4x)dx (b) e 5x sin(3x)dx (c) e x + 2e x dx
7 (d) 6x x2 dx (e) 3x 3 + x x + 3 x 2 dx + 9 (f) cos 9 xsin 3 xdx
8 Exmple. Clculte Section 7.4 Algebric Identities 2 (x )(x + ) dx. Exmple 2. For ech of the following, find the form of the prtil frctions decomposition. Do NOT solve for the constnts. () 0 x 2 x 6 (b) 3 2x (x 2 6x + 9)(x 2 + ) (c) 5x 3 2 (x 5 x 3 )(x 2 + 4) 2
9 Informtion on Prtil Frctions Prtil Frctions for Integrting Rtionl Functions. where P(x) nd Q(x) re polynomils. P(x) Q(x) dx, Cse. P(x) hs lower degree thn Q(x). In this cse, formulte your prtil frctions decomposition s suggested below. Type of Fctor in Q(x) Corresponding Term(s) in Prtil Frction Decomposition A Liner: x c x c Repeted Liner: (x c) n A x c + A 2 (x c) A n (x c) n Irreducible Ax + B Qudrtic: q(x) q(x) Repeted Irreducible Qudrtic: q(x) n A x + B + A 2x + B 2 q(x) q(x) A kx + B k q(x) n Cse 2. P(x) hs equl or greter degree thn Q(x). In this cse, long division must be used first, nd then Cse must be pplied to the reminder term. Exercises. Clculte ech of the following: () (b) 2x 2 + 4x 9 x 3 dx + 9x 4x 2 7x + x(2x ) 2 dx
10 Section 7.7 & 7.8 Improper Integrls Preliminry Exmple. To the right, you re given the grphs of y = /x 2 nd y = /x. Clculte the improper integrls x dx. dx nd x2 Figure. Grph of y = /x 2 Figure 2. Grph of y = /x Types of Improper Integrls Improper Integrl Definition Picture f(x)dx = b f(x)dx = b
11 Improper Integrl Definition Picture f(x)dx = b f(x)dx = b b f(x)dx = b b f(x)dx = c b Definition. We sy tht n improper integrl if ll of the defining limits exist nd re finite. If ny of the involved limits is infinite or does not exist, we sy tht the improper integrl. Comprison Theorem. Suppose tht f nd g re continuous functions with f(x) g(x) 0 for x.. If is convergent, then is convergent. 2. If is divergent, then is divergent. g(x) f(x)
12 Exmple. Consider the integrl 3 x 4 dx. () Explin why the bove integrl is improper. (b) Determine whether or not this integrl converges. If it does converge, find its vlue. Exmple 2. Determine whether or not 0 xlnxdx converges. If it does converge, find its vlue.
13 Exmples nd Exercises. Determine whether or not lnx dx converges. If it does converge, find its vlue. x2 Theorem. Let k be positive rel number. Then (I) The integrl dx converges if p > but diverges if p. xp (II) The integrl k k dx converges if m > 0 but diverges if m 0. emx 2. Use the Comprison Theorem to determine whether or not the integrl x 4 dx converges Use the Comprison Theorem to nswer ech of the following questions. () Does (b) Does (c) Does 2 2 x dx converge or diverge? dx converge or diverge? x + x6 e x2 dx converge or diverge? 4. Is the solution to the problem below correct or incorrect? Explin. Problem. Determine whether dx converges or diverges. x + Solution. Since for ll x, the denomintor of x. This mens tht Therefore, since lso diverge by the Comprison Theorem. x + x for ll x. x + is bigger thn the denomintor of x dx diverges by fct (i) from the previous pge, 2 x + dx must
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