6/8 of a gallon of orange juice is left. Six thirsty people want an equal amount of orange juice. How much does each person get?
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Division yields
Now for partial fractions: you're looking for constants <em>a</em>, <em>b</em>, and <em>c</em> such that
which gives <em>a</em> + <em>b</em> = 2, <em>c</em> = 0, and 2<em>a</em> = -7, so that <em>a</em> = -7/2 and <em>b</em> = 11/2. Then
Now, in the integral we get
The first two terms are trivial to integrate. For the third, substitute <em>y</em> = <em>x</em> ² + 2 and d<em>y</em> = 2<em>x</em> d<em>x</em> to get