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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 Fall '10
 KIHYUNHYUN
 Calculus, Derivative, Fundamental Theorem Of Calculus, dx

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