Problem 45 Hard Difficulty

Use the table of integrals to evaluate $ F(x) = \int f(x)\ dx $, where
$$ f(x) = \frac{1}{x \sqrt{1 - x^2}} $$
What is the domain of $ f $ and $ F $?
(b) Use a $ CAS $ to evaluate $ F(x) $. What is the domain of the function $ F $ that the $ CAS $ produces? Is there a discrepancy between this domain and the domain of the function $ F $ that you found in part (a)?

Answer

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Video Transcript

Okay, So this question wants us to find the anti derivative of this function in two ways. One with the table and another using a computer algebra system. So the first single wants us to do is use are integral table. So using the formula from the table plugging in a equals one, we're given this anti derivative so part, eh wants the domain of F, which is the function we just found the anti derivative us. So what is the domain of f of X equals one over X square root one minus x squared. So first of all, ex cannot equal zero, or else our denominator will equal infinity and also ex con are equal one. Because then our denominator would equal zero again. And then we got to consider the square root term. So one minus X squared has to be a positive number. So that means that X squared has to be less than one. So that means that X must be absolute value of X. That is must be less than one. So that's our domain. So now it wants the domain of capital F. So capital f is this function, and it's exactly the same cause. We are dividing by an ex somewhere, and we have a one mind sex squared in the skirt. So the domain of F is the same thing. So now that we have our domains and if you want to write them in set notation F and capital ask both have domains of negative 10 Union 01 Okay, Now it asks us to do the same thing with a computer algebra system derivative. So we're gonna call this F sub to Lex because it's the second insider of forgiven, and it's actually the exact same. So the domain is still negative. 1 to 0. Union 01 Okay. And this might vary depending on what computer algebra system you use. The one I used was just medical calculator, and it gave the exact form of the table. But others may restrict your domain based on what functions they using the anti derivative. So it could be stuff like hyperbolic in verses or something. But ours happened to work out, so that was good