ACTIVITY NO. 6 (Combining Lessons No. 7 and 8) Evaluate the following: 1. f cos 7x cos 4x dx 3. [ sin*xcos3 xdx 2. [ cos - sin 4x dx 4. [ sin+3x dx 5LESSON 7: TECHNIQUES OF INTEGRATION - PRODUCTS OF SINES and COSINES TECHNIQUES OF INTEGRATION Manipulations that make integration easier; Converts the integrand into something that can be solved using previously given theorems II. PRODUCTS of SINES and COSINES T43] [ sinu sinv dx = = ][cos(u - v) - cos(u + v)]dx T44] S sinu cosv dx = = f[sin(u - v) + sin(u + v)]dx T45] [ cosu cosv dx = ; ][cos(u - v) + cos(u + v)]dx Where: u,v = functions of x, uz vLESSON 8: TECHNIQUES OF INTEGRATION - POWERS OF SINES and COSINES III. POWERS of SINES and COSINES (Iihim 1. Evaluate integrals involving powers of sine and cosine functions 2. Solve integration powers using previously learned trigonometric identities 3. Convert integrand into something that can be solved using previously discussed theorems 4. Make integration of complicated differentials easier General Form: j'sinmvcosnvdx Where: v = function ofx m, n = whole numbers Note: Do not interchange m and n; m is the exponent of the sine function, n is the exponent of the cosine function T46] Case 1. m is a positive odd integerI n is any number I sinmvcosnvdx = f sinm'lvcosnvsin v dx Then use the identityr sinz = 1 c0328 T47] Case 2. n is a positive odd integer, m is any number [ sinmvcos"vax = [ sinmvcos"-lycos vdx Then use the identity cos20 = 1 - sin20 T48] Case 3. m and n are both positive, or one is positive, and the other is zero m n [ sinvcos vax = S(sin v)2(cos v)idx Then use one or both of the following identities: sin20 = 1-cos 20 cos20 = 1+cos 20 2 2