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The sine integral function $$ \displaystyle \text{Si}(x) = \int^x_0 \frac{\sin t}{t} \, dt $$ is important in electrical engineering. [The integrand $ f(t) = (\sin t)/t $ is not defined when $ t = 0 $, but we know that its limit is 1 when $ t \to 0 $. So we defined $ f(0) = 1 $ and this makes $ f $ a continuous function everywhere.]

(a) Draw the graph of $ \text{Si} $.(b) At what values of $ x $ does this function have local maximum values?(c) Find the coordinates of the first inflection point to the right of the origin.(d) Does this function have horizontal asymptotes?(e) Solve the following equation correct to one decimal place: $$ \displaystyle \int^x_0 \frac{\sin t}{t} \, dt = 1 $$

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Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Integration

Missouri State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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The sine integral function…

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(a) The sine integral func…

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In calculus, the definite …

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someplace we have a scientist or a function and I'm starting And the problem asked. Explain a bit about how you defy zero because it's not defined but we can use the limit value to define functions From here continues I'LL leave the drawing part two You may use a graphic calculator and howto figure out no home local maximum local maximum We want we want the first derivative which is science over X Toby to be zero That's a cripple point. So here, that gives us ex Pecos K pi case in future Okay, cannot be zero bills, by our definition have zero because what by By using the limit value and and if you're right, I guess is fine. And, uh and we also want local maximum. So we are also on the checks are seven derivative is a negative. So there's computer formula for seventy roof to you Wish by kosher rule gives us this So we also need this Toby A negative. So here, Bessie Just have the check, Sir, If you probably in Cape Hi thiss parties always zero. So basically this is basically a sigh off Exco Sigh Ox So I mean, there were quickly flee with this art When case Even Cho size positive one. Yeah, Wei wants to be next. So so here. That tells us that kay, if case even this is positive, we want this to detective. So a case either even and, uh, active or art and the positive. So those are OK that will make this formula start this for this formula next event that gives us a local man local maximum. So actually, it was Cape High on case case thes passport. It's integers first, the inflection point. That's right with the origin. So it's open another page I'm going to solve ex coz i X minus Science inflection point seven derivative will be zero So that keeps us. So when the ship we want this to be zero No. So on the salty sea Air Force Zero. So you have science over co science because x so that is solving tension X you know, sex. It's why you don't have and the formula. But so you get asked estimated by using the graphic calculator. No, and I would leave that part to you. Does this function have horizontal ass in tow? So but ask goes to infinity it as this convert basically on that pie over too. But your affinity No, here should be X our society over so horizontal s until basically asked you if this it's gross does it is suitable converge. I'm not sure if there's could wait to it, maybe even use a graphic, calculate her to see a craft and to make a conclusion. But but the answer's yes. And you'LL know howto computers in the girl for Maura events class. If I remember correctly the answer of this this limits should be pi over too. But you're not going to learn thiss class you got If you're a math major in my everything more advanced class o r for the purpose of this probably just use of graphic calculator to to see it and I believe the last part for you to buy using the calculator

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