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Math 1432 Section 15251 Aarti Jajoo [email protected] (add Math 1432 in subject) 606 PGH Office Hours in 606 PGH: Starting from Wednesday Monday 1:30 – 2:30 p.m. Wednesday 1:30 – 2:30 p.m. POPPER 1. x ln x 2 dx = ∫ a. x 2 ln x + C x2 b. − +C 2 x2 c. x ln x + +C 2 2 x d. x 2 ln x − +C 2 e. None of these 2. ∫ arctan xdx 3. ∫ x 2 arctan xdx 4. ∫x 3 − x2 e dx Integration by parts with definite integrals. ∫ 5. ∫ 1 0 xe x dx b a b u dv = ( uv )| − a ∫ b a v du 6. π2 ∫0 x 2 sin x dx 7. ∫( e x + 2x ) 2 dx Section 8.3 Powers and Products of Trigonometric Functions Recall the following identities: cos 2 ( x ) + sin 2 ( x ) = 1 1 + tan 2 ( x ) = sec 2 ( x ) 1 + cot 2 ( x ) = csc 2 ( x ) cos 2 (x) = 1 + cos ( 2 x ) 2 sin ( 2 x ) = 2 sin x cos x cos ( 2 x ) = cos 2 x − sin 2 x sin 2 (x) = 1 − cos ( 2 x ) 2 In this section, we will study techniques for evaluating integrals of the form ∫ m n sin x cos x dx and ∫ m n sec x tan dx where either m or n is a positive integer. To find antiderivatives for these forms, try to break them into combinations of trigonometric integrals to which you can apply the Power Rule, which is ∫ ⎧ u n+1 +C ⎪ n u du = ⎨ n + 1 ⎪ln u + C ⎩ if n ≠ 1 if n = − 1 . Integrals Involving Powers of Sine and Cosine 1. If the power of the sine is odd and positive, save one sine factor to be the du and convert the remaining factors to cosine. Then, expand and integrate. ∫ sin 3 x dx ∫ sin 3 x cos 2 x dx 2. If the power of the cosine is odd and positive, save one cosine factor to be the du and convert the remaining factors to sine Then, expand and integrate. ∫ cos 5 x dx ∫ sin 4 x cos 5 x dx 3. If the powers of both the sine and cosine are even and nonnegative, make repeated use of the half angle identities: sin 2 x = ∫ cos 2 x dx 1 − cos 2 x 2 and cos 2 x = 1 + cos 2x . 2 ∫ cos 4 x dx 4 2 cos x sin x dx ∫ Integrals involving Secants and Tangents 1. If the power of the secant is even and positive, save a secant squared factor to be the du and convert the remaining factors to tangents. Then, expand and integrate. ∫ sec 4 x dx ∫ sec 4 x tan x dx 2. If the power of the tangent is odd and positive, and there is a secant factor, save a secant-tangent factor to be the du and convert the remaining factors to secants. Then, expand and integrate. ∫ tan 3 x sec x dx Compute 3 sec 2x tan ( ) ( 2x ) dx ∫ If there is no secant factor, convert a tangent squared factor to a secant squared factor; then expand and repeat if necessary. ∫ tan 3 x dx 3. If there are no secant factors and the power of the tangent is even and positive, convert a tangent squared factor to a secant squared factor; then expand and repeat if necessary. ∫ tan 4 x dx m sec x dx where m is odd and ∫ positive, use integration by parts. 4. If the integral is of the form ∫ sec 3 x dx 5. If none of the first four cases applies, try converting to sines and cosines. sec x ∫ tan 2 x dx Integrals involving Sine-Cosine Products with Different Angles Use the product-to-sum identities. 1 sin A sin B = cos ( A − B ) − cos ( A + B ) 2 ( 1 sin A cos B = sin ( A − B ) + sin ( A + B ) 2 ( ) 1 cos A cos B = cos ( A − B ) + cos ( A + B ) 2 ( ) ) Rewrite sin (5x) sin (3x) ∫ sin ( 5 x ) sin ( 3 x ) dx =