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X-3-2-10123f(x)14204112458how would you go about finding the antiderivative from a tableof values, like this, without having a given value of F(x)?...

Question

X-3-2-10123f(x)14204112458how would you go about finding the antiderivative from a tableof values, like this, without having a given value of F(x)?

x -3 -2 -1 0 1 2 3 f(x) 14 20 4 11 24 5 8 how would you go about finding the antiderivative from a table of values, like this, without having a given value of F(x)?



Answers

$23-24$ Find the antiderivative $F$ of $f$ that satisfies the given condition. Check your answer by comparing the graphs of $f$ and $F .$
$$f(x)=4-3\left(1+x^{2}\right)^{-1}, \quad F(1)=0$$

We're asked to find the anti derivative of this function defined as For X. to the 3/2ves -1. All over X. Now it's implied that that's X to the first power because of what I would do is rewrite this Um and separate those. So we're looking at -1 over X. So what we're really looking at is finding the anti derivative of when you divide with the same base Right here, you subtract those exponents. Well, three has -1 would give me X to the one half power. Now I wouldn't mess with this one over X. Um Like you could rewrite that as X to the negative first power. Um But the Power rule does not apply to that one. So when you do the when you find the anti derivative, yes, you do add one to your exponent where you can and then you multiply by the reciprocal of four times two thirds will give me eight thirds and double check. That's right. Because the derivative of capital left needs the equal lower case F. Uh And there is no I didn't mess with the one over X. Is that is equal to natural log of the absolute value of X. And don't forget about your plus C. So again, I think I've explain this sufficiently that the derivative of capital F will work backwards to give you lower case F. To verify that we have the right answer which we should.

We want to find the anti derivative of function F of X is equal to two extra three halfs minus three X. This question is testing our knowledge of anti differentiation as a precursor of integration. So we need to understand this concept. Remember that the anti derivative is just the inverse derivative. So if we understand derivative rules, we should understand how to find an anti derivative for this problem to actually three half minutes, three acts. We only need rule one out of the three rules they have listed here. That is the power rule. Since the power rule states that DDX extra is a actually a -1. It holds by the inverse. That anti derivative actually A is 1/8 plus one times actually eight plus one. So we must have that anti derivative capital F X is two times two X. The five half minus three times one half X squared. This is implementing one over a plus one times actually a plus one. As I mentioned, multiplying by our Constance two and negative three respectively. That's we have final solution. Capital F x is 4/5 extra five half minus three half X squared.

We want to find the anti director of the function F of X equals four, expose three to the one half. This question is has to get dollars or anti differentiation as a precursor to integration. Uh Such when you understand this concept to finish this question, remember that an anti derivative is just an inverse derivative function. What that means is that if you understand the rules for taking derivatives, you understand how to take an anti derivative by the members for this problem. We need to use both. The power rule of the chain rule. The reason we could use the chain rule as a highlight it is because we take the derivative dysfunction four X plus three to the one half. It would produce a factor of four by the chain rule. Therefore when we take the inverse chain rule, it must be that anti driving this absorbs a factor of four. That is we divide by four. So we proceed the anti derive with this information in hand. At access two thirds four expose three to the three house by the power rule, times 1/4 by the chain rule. Therefore, we have the anti derivative is capital. F. X equals +164 expose three to the three has.

We wish to evaluate this integral here using the table gift. So let's start with integration by parts. We're going to let you be equal to X and D. V is going to be a prime of Agostina. Now, do you? That's just X and B will be the anti derivative ah F prime of X. So that would just be Yeah. And now let's go ahead and put this back in. So this is going to be equal to few times to be. So that would be X times after effects. And we evaluate that from 0 to 5. Subtract The integral from 0 to 5 of google. So that would be if the legs the X. Okay, so this is equal to let's just copy the first part down. Okay, so now the anti derivative of lower case F a bex. That is just capital of Quebec's because we're given that lower case alphabet is capital F. Trying to vex. So this is really just half of X Evaluated from 0 to And the table has given us all of these values. So this would be five times out of five minus zero times up of zero and then minus F of five -F of zero. Okay, so this is five times and lower case five is 27. This is just zero and minus Capital after five is 20. Capital F, zero is 10. So this is five times 27, which is 1:35 -10. So our answer is 120.


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