june-1-07-ans

# june-1-07-ans - MAC 2312 Answers to Test 1 given on June 1...

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Unformatted text preview: MAC 2312 Answers to Test 1 given on June 1, 2007 In 1–10, find each integral. 1. Z sin 2 4 x dx Use the half angle formula sin 2 θ = 1 2 (1- cos2 θ ) Z 1 2 (1- cos8 x ) dx = 1 2 Z 1 dx- 1 8 Z 4cos8 xdx = 1 2 x- 1 8 sin8 x + C 2. Z tan 4 x sec 6 x dx Use trig identity sec 2 x = 1 + tan 2 x . Use integration formula Z udu = u n +1 n + 1 . Z tan 4 x sec 4 x sec 2 xdx = Z tan 4 x (sec 2 x ) 2 sec 2 x dx = Z tan 4 x (1 + tan 2 x )2sec 2 xdx Subsituting u = tan x du = sec 2 x dx Z u 4 (1 + u 2 ) 2 du = Z u 4 (1 + 2 u 2 + u 4 ) du = Z ( u 4 + 2 u 6 + u 8 ) du = Z u 4 du + 2 Z u 6 du + Z u 8 du = 1 5 u 5 + 2 7 u 7 + 1 9 u 2 + C = 1 5 tan 5 x + 2 7 tan 7 x + 1 9 tan 9 x + C 3. Z e 4 x dx √ 9- e 8 x Use integration formula Z du √ a 2- u 2 = Arcsin u a + C Z e 4 x dx q (3) 2- ( e 4 x ) 2 = 1 4 Z e 4 x 4 dx q (3) 2- ( e 4 x ) 2 = 1 4 Arcsin e 4 x 3 + C 4. Z e 8 x dx √ 9- e 8 x Use integration formula Z u n du = u n +1 n + 1 + C Z (9- e 8 x )- 1 2 e 8 x dx =- 1 8 Z (9- e 8 x )- 1 2 e 8 x 8...
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june-1-07-ans - MAC 2312 Answers to Test 1 given on June 1...

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