Chapter 7 - Review Chapter 7(Section 7.1 7.5 Section 7.1...

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Review : Chapter 7(Section 7.1 - 7.5) Section 7.1 Integration by Parts Formula for Integration by Parts i f ( x ) g ( x ) dx = f ( x ) g ( x ) i g ( x ) f ( x ) dx ⇐⇒ i u dv = uv i v du where u = f ( x ), v = g ( x ), and the diferentials du = f ( x ) dx and dv = g ( x ) dx . De±nite Integrals by Parts i b a f ( x ) g ( x ) dx = f ( x ) g ( x )] b a i b a g ( x ) f ( x ) dx Problem 1. Evaluate the integrals. (a) i π/ 2 0 x 2 cos 2 x dx (b) i e 2 y cos 2 y dy (c) i ( r 2 + r + 1) e r dr (d) i x sec 1 x dx (e) i 2 1 x ln x dx (F) i ln ( x + x 2 ) dx Section 7.2 Trigonometric Integrals Usefule Formulae and Identities for Trigonometry 1. sin 2 x = 1 cos (2 x ) 2 cos 2 x = 1 + cos (2 x ) 2 2. sin (2 x ) = 2 sin x cos x cos (2 x ) = cos 2 x sin 2 x 3. sin 2 x + cos 2 x = 1 , 1 + tan 2 x = sec 2 x, 1 + cot 2 x = csc 2 x Strategy for Evaluating i sin m x cos n x dx 1. IF n = 2 k + 1(odd), save one cosine Factor and use cos 2 x = 1 sin 2 x to express the remaining Factors in terms oF sine: i sin m x cos 2 k +1 x dx = i sin m x (cos 2 x ) k cos x dx = i sin m x (1 sin 2 x ) k cos x dx 1
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Then substitute u = sin x , and du = cos xdx . 2. If m = 2 k + 1(odd), save one sine factor and use sin 2 x = 1 cos 2 x to express the remaining factors in terms of cosine: i sin 2 k +1 x cos n x dx = i (sin 2 x ) k cos n x sin x dx = i (1 cos 2 x ) k cos n x sin x dx Then substitute u = cos x , and du = sin xdx ⇔ − du = sin xdx . 3. If both m and n are even, use the half-angle identities sin 2 x = 1 2 (1 cos 2 x ) cos 2 x = 1 2 (1 + cos 2 x ) It is sometimes helpful to use the identity sin x cos x = 1
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