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Math Help - Calculating a Definite Integral

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    Junior Member ReneG's Avatar
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    Calculating a Definite Integral

    Hello, I'm a freshman in highschool trying to get a headstart into calculus, so please excuse my limited knowledge in the subject.

    Anyways, I have this integral

    \int_2^8{\left(\frac{x}{8} - \frac{2}{x^2}\right)} dx

    I have some sort of idea on how to solve it. Something about taking the anti-derivative.

    The answer is 3, but I would like to know how to get there step by step, I'm having a hard time comprehending my calculus book.
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    Re: Calculating a Definite Integral

    If \displaystyle F(x) is an antiderivative of \displaystyle f(x), then the area under the curve between x = a and x = b is given by \displaystyle \int_a^b{f(x)\,dx} = F(b) - F(a). In other words, evaluate an antiderivative of your function, evaluate it at x = b and x = a, and then evaluate the difference between these values.
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    Junior Member ReneG's Avatar
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    Re: Calculating a Definite Integral

    How would I find the antiderivative of \frac{x}{8} - \frac{2}{x^2} though?

    Unfortunately, I've only learned the Power Rule and its counter part. Even then it was just one term.

    Thanks for your reply.
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    Re: Calculating a Definite Integral

    The antiderivative of a sum/difference is equal to the sum/difference of antiderivatives. So you can work out the antiderivative of each term.

    It would help if you rewrote your function as \displaystyle \frac{1}{8}x - 2x^{-2}.
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    Junior Member ReneG's Avatar
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    Re: Calculating a Definite Integral

    Ahh, I see it now. Would the antiderivative of the aforementioned function be \frac{x^2}{16} + \frac{2}{x} ? I know I could just check if the difference of F(8) - F(2) is 3 to check, but I just want to make sure I'm doing this right.
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    Re: Calculating a Definite Integral

    Quote Originally Posted by ReneG View Post
    Ahh, I see it now. Would the antiderivative of the aforementioned function be \frac{x^2}{16} + \frac{2}{x} ? I know I could just check if the difference of F(8) - F(2) is 3 to check, but I just want to make sure I'm doing this right.
    That is correct.
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