4.7.7-eg - Example Evaluate the indefinite integral √ 3...

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Unformatted text preview: Example Evaluate the indefinite integral √ 3 x − 2x 3 x dx. Solution: Separating the integrand at the minus sign in the numerator, √ √ 3 x − 2x3 3 x 2x3 dx = − dx x x x = 3x−1/2 − 2x2 dx =3 x−1/2 dx − 2 (1) x2 dx, where in the last equality we have used the linearity property of the integral, namely that (af (x) + bg (x)) dx = a f (x) dx + b g (x) dx, for constants a and b. It then follows that √ x1/2 x3 3 x − 2x3 dx = 3 − 2 + C, x 1/2 3 2x3 = 6x1/2 − + C. 3 where we have combined the two constants of integration into the one arbitrary constant C . Note: The answer can always be checked by differentiating the answer given above and confirming that the integrand is recovered. That is, d 2x3 6x1/2 − +C dx 3 1 2 = 6 · x−1/2 − · 3x2 = 3x−1/2 − 2x2 , 2 3 which is the separated form of the integrand given in (1) above. ...
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