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# Mock Gateway 4 Answers - π 3-= π 3 8 3 = e ln 3 3 x =...

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Answers to Mock Gateway IV 1. f ( x ) = e x + cos x + C f (0) = 2 + C C = - 1 f ( x ) = e x + cos x - 1 2. s ( t ) = - t + C s (0) = C C = 3 s ( t ) = - t + 3 s ( t ) = - 1 2 t 2 + 3 t + D s (0) = D D = 7 s ( t ) = - 1 2 t 2 + 3 t + 7 3. 1 0 ( x 2 - 2 x + 2) dx = 1 3 x 3 - x 2 + 2 x 1 0 = 1 3 (1) 3 - (1) 2 + 2(1) - 1 3 (0) 3 - (0) 2 + 2(0) = 4 3 4. 1 x 2 - x + 2 x dx = ( x - 2 - x 1 / 2 + 2 x - 1 / 2 ) dx = x - 1 - 2 3 x 3 / 2 + 4 x 1 / 2 + C

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5. e 1 3 x dx = 3 e 1 1 x dx = 3 ln x e 1 = 3 ln e - 3 ln 1 = 3 6. π/ 2 0 2 cos θdθ = 2 sin θ π/ 2 0 = 2 sin π/ 2 - 2 sin 0 = 2 7. . (a) 1 x 2 + 1 dx = Tan - 1 x + C (b) 3 / 2 0 1 1 - x 2 dx = Sin - 1 x 3 / 2 0 = Sin -
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Unformatted text preview: π 3-= π 3 8. 3 = e ln 3 3 x = e (ln 3) x Z 3 x dx = Z e (ln 3) x dx Substitute u = (ln 3) x . Then du = (ln 3) dx and we get Z e u ± 1 ln 3 ² du = e u ln 3 + C = e (ln 3) x ln 3 + C = 3 x ln 3 + C 9. F ( x ) = e-x 2...
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