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

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

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.

Re: Calculating a Definite Integral

If $\displaystyle \displaystyle F(x)$ is an antiderivative of $\displaystyle \displaystyle f(x)$, then the area under the curve between x = a and x = b is given by $\displaystyle \displaystyle \int_a^b{f(x)\,dx} = F(b) - F(a)$. 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.

Re: Calculating a Definite Integral

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

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

Thanks for your reply.

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 $\displaystyle \displaystyle \frac{1}{8}x - 2x^{-2}$.

Re: Calculating a Definite Integral

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

Re: Calculating a Definite Integral

Quote:

Originally Posted by

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

That is correct.