How to calculate this function using an integral and derivative?

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Gissa

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How would you calculate this function? When I try to look it up online it only gives me answers with Si(t)/3 but it is given that the answer is the top left one. (See attachments)

1705446546097.png
1705446565081.png
 
When trying to search for the answer it only gives me the answer: Si(x)/3 where Si is the derivative of sin. But the format of the answers are given (see attachment)
The formula is:1705488542361.png
The answer is:1705488611595.png
 
Gissa said:
When trying to search for the answer it only gives me the answer: Si(x)/3 where Si is the derivative of sin. But the format of the answers are given (see attachment)
The formula is:View attachment 36936
The answer is:View attachment 36937
Click to expand...

What is the "it" that "gives" you "Si(x)/3"?

What have *you* done about doing the integration, evaluation, and then differentiation?
 
Gissa said:
When trying to search for the answer it only gives me the answer: Si(x)/3 where Si is the derivative of sin. But the format of the answers are given (see attachment)
The formula is:View attachment 36936
The answer is:View attachment 36937
Click to expand...
I take it you asked something like Wolfram Alpha for the indefinite integral. But Si(x) is not the derivative of the sine, it's the "sine integral", which is just a name for that very integral. It doesn't help you at all.

This problem doesn't require doing that integration. It is presumably an exercise given after you have learned the Fundamental Theorem of Calculus. Use that theorem:

math.libretexts.org

5.3: The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this …
math.libretexts.org math.libretexts.org

See Part I, and combine it with the chain rule.
 
Dr.Peterson said:
I take it you asked something like Wolfram Alpha for the indefinite integral. But Si(x) is not the derivative of the sine, it's the "sine integral", which is just a name for that very integral. It doesn't help you at all.

This problem doesn't require doing that integration. It is presumably an exercise given after you have learned the Fundamental Theorem of Calculus. Use that theorem:

math.libretexts.org

5.3: The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this …
math.libretexts.org math.libretexts.org

See Part I, and combine it with the chain rule.
Click to expand...
Ah thank you so much, I missed a lecture so I guess that I missed the 'Fundamental theorem of calculus'. I will try to solve it after watching some videos about that theorem. And yes I asked wolfram alpha for the answer which was not really helpful.
 
Gissa said:
Ah thank you so much, I missed a lecture so I guess that I missed the 'Fundamental theorem of calculus'. I will try to solve it after watching some videos about that theorem. And yes I asked wolfram alpha for the answer which was not really helpful.
Click to expand...
If you asked it the entire question, it would have given a correct answer (though you'd have to look up something more in order to understand it).

But that's not how to learn! What it tells you would not teach you how to do the work, or even mention the FTC, which is presumably what you are supposed to be learning. (Have you tried reading your textbook, or whatever other resources the lecturer assumes you have? Or even the textbook I gave you a link to?)
 
I have read the textbook you linked in the previous response and was able to solve the question. Now that I know how to use the FTC it is not that hard of a problem anymore haha, thank you.
 
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