tv2-1-06-ans - MAC 2312 Date February 1 2006 Name(Print...

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MAC 2312 Date: February 1, 2006 Test 1 Name (Print) Please enclose the simplified answer in a box . Please print your name on each page. No Books. No Notes. No Calculators. Find each integral. 1. Z sin 4 x cos 5 x dx . Use the integration formula Z u 2 du = u n +1 n + 1 + C . Z sin 4 x cos 5 x dx = sin 4 x cos 4 x cos c dx = sin 4 x (cos 2 x ) 2 cos x dx Use the trig identity cos 2 x = 1 = sin 2 x . = Z sin 4 x (1 - sin 2 x ) cos x dx u = sin x du = cos x dx = Z u 4 (1 - u 2 ) du = Z u 4 (1 - 2 u 2 + u 4 ) 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 9 + C = 1 5 sin 5 x - 2 7 sin 7 x + 1 9 sin 9 x + C 2. (a) Z x 3 dx 4 - x 4 Use the integration formula Z u n du = u n +1 n + 1 . Z x 3 dx 4 - x 4 = Z (4 - x 4 ) - 1 2 x 3 dx u = 4 - x 4 du = - 4 x 3 dx = - 1 4 Z (4 - x 4 ) - 1 2 4 x 3 dx = - 1 4 · 2 (4 - x 4 ) 1 2 = - 1 2 (4 - x 4 ) 1 2 + C
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(b) Z x dx 4 - x 4 Use the integration formula Z du a 2 - u 2 = arcsin u a + C Z x dx 4 - x 4 = Z x dx q 2 2 - ( x 2 ) 2 u = x 2 du = 2 x dx = 1 2 Z 2 x dx q 2 2 - ( x 2 ) 2 = 1 2 arcsin x 2 2 + C 3.
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