4.7.2-eg-1 - Example Find the antiderivative F (x) of f (x)...

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Unformatted text preview: Example Find the antiderivative F (x) of f (x) = 2 x4 − 9 2x 2 which satisfies F (−1) = 10. Solution: Since 9 f (x) = 2x−4 − x−2 , 2 it follows from the linearity of antiderivatives and the fact that an antiderivative of xn is xn+1 given by when n = −1, that the most general antiderivative F (x) of f (x) is given by n+1 F ( x) = 2 · x− 3 9 x− 1 2 9 −· +C =− 3 + + C, −3 2 −1 3x 2x where C is an arbitrary constant. If F must also satisfies F (−1) = 10, then 10 = F (−1) = − 2 9 29 + + C = − + C, 3 3(−1) 2(−1) 32 and it follows that C = 10 − Thus F (x) = − 29 83 +=. 32 6 2 9 83 + +. 3x3 2x 6 Note: The answer can always be checked by confirming that F (x) = d 2 9 83 2 9 − 3+ = 4 − 2 = f (x), + dx 3x 2x 6 x 2x and F (−1) = − 2 9 83 2 9 83 + + =−+ = 10. 3 3(−1) 2(−1) 6 32 6 ...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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