(cos 3x sin 3x)(cos 3x  (cos 3x sin 3x) 3x)dx (cos 3x sin 3x)dx  sin 3x = cos 3x 3  c 3 5.6 Integration using the identities, 2 sin A cos B = sin (A + B) + sin (A B) – 2 cos A cos B = cos (A + B) + cos (A B) – -2 sin A sin B = cos (A+B) cos (A B) – – These identities convert the product into sums. The following examples illustrate the use of these identities. Example 8 ∫sin Find 10x cos8x dx Solution Using the identity 2 sin A cos B = sin (A + B) + sin (A B) – 1 We have sin 10x cos 8xdx = 2 sin 10x cos 8xdx = 1 (sin18x sin 2x)dx  1 1 1 2 18 2 = 36 4 Example 9 Find cos 12 sin 3 x xdx Solution cos 12 sin 3 sin 3 cos 12 x xdx x xdx = 12 xdx 2 = 3 c x 1 46 = =
(cos 3x  sin 3x)(cos 3x 3x)dx (cos 3x sin 3x) (cos 3x  sin 3x)dx = sin 3x 3 cos 3x 3  c 5.6 Integration using the identities, 2 sin A cos B = sin (A + B) + sin (A – B) 2 cos A cos B = cos (A + B) + cos (A – B) -2 sin A sin B = cos (A+B) – cos (A – B) These identities convert the product into sums. The following examples illustrate the use of these identities. Example 8 Find ∫sin 10x cos8x dx Solution Using the identity 2 sin A cos B = sin (A + B) + sin (A – B) We have sin 10x cos 8xdx = 1 2 sin 10x cos 8xdx = 1 (sin18x  sin 2x)dx 1 1 1 2 18 2 = 36 4 Example 9 Find cos 12 x sin 3 xdx Solution cos 12 x sin 3 xdx = sin 3 x cos 12 xdx = 1 2 3 x 12 xdx 46 = = c
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