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Of the Fourier transform (FT) X(f) of time domain signal 1, Write down the definition x(t) € € where t € R...

Question

Of the Fourier transform (FT) X(f) of time domain signal 1, Write down the definition x(t) € € where t € R

of the Fourier transform (FT) X(f) of time domain signal 1, Write down the definition x(t) € € where t € R



Answers

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
g(x) =
x
1
t3 + 3
dt

1

In this problem we have to find value of lap loss of the function scientology upon T. So here we can see that our function scient E is divided by T. So from the rule of division by T. We know that if any function is divided by T. Let us say F F P is divided by T. And we have to find lap lots of this function. Then this can be written as integration off F F S D. S from S to infinity where ffs how we will find fFS. So this is this FFS is basically lap loss of that function which is divided by this is a divided by T. So laplace off F F T is equals two F F. S. So here first we have to identify which function is FFT and what is F of S. So as we can see that scientist is being divided by T. Therefore in this case FFT is 70. Therefore lap lots of fft will be equals two lap plus of scientists, one upon S E squared plus one. Therefore, this dysfunction becomes capital FFS. Therefore now we can put the Now we have got the function of office so now we can integrate here. Therefore lap plus off 70 divided by T will be equals to integration from S to infinity. What is fs It is one upon as a scar plus one. So we can write it as one plus S. A square and we have to integrate with respect to S. Now recall the basic fundamental of integration. Integration of one plus X squared dx is 10 in verse six. Therefore here instead of X. The variable as therefore integration of one places square will be 10 universe. As now we have to put the limit from S to infinity. So we have to evaluate this, evaluate this function under these limits. So if you put the upper limit, so 10 inverse infinity, we know that value of 10, 5 by two is infinity. Therefore 10 inverse infinity will give by by two. So this is the value of the function at upper limit. Now we have to subtract the lower limit and that is 10 verse is therefore lap lots of 70 divided by T will be equals two five to minus dan universe. Yes. So this will be there answer for this problem.

Mm hmm. Okay. Here I have four integral. Then I'm gonna integrate for you. I have X to the second D. X. So when I integrate that I'm going to get a function of X. I get X to the third over three plus some constant of integration. Now if it's T square D. T. Then I'm going to integrate with respect to T. But I still use the same rule which is add one and divide. So T cubed over three plus C. If it's P squared OK. Same rule be cubed over three plus C. Or if it's triangle or if it's a flower if it's hard, whatever it is. Okay. Same rule, that thing cubed over three plus C. So you can see that it doesn't matter what letter I use in there or symbol. I still integrate it the same way. So it's really not going to make a difference when you're doing definite into girls. Like when you're integrating from let's say 0 to 6 X squared dx you're gonna get X to the third over three from 0 to 6. So you're gonna get six cubed over three minus zero cubed over three. So that's 26 72. But if I did 0 to 60 square D. T. I'm gonna get exactly the same answer. Okay? Because the the name of the thing that you're integrating doesn't matter. That's what they mean when they say dummy variable they just mean it's sort of a placeholder and it doesn't matter if you use X. Or T. Or P. Or P. Or whatever. You'll get the same number when you're doing um definite integration or the same answer. Just with a different letter in it. If you're doing indefinite integration. So if you have D. X. You're gonna have X. And your answer. If you have t. You're gonna have tea and your answer you have p. You're gonna pee in your answer. But if you have numbers on the on the integral sign then you're going to have a number answer. Okay? If that wasn't what you wanted to know, just ask your question again.

In this problem we are given with the function G. S equals to integration of one party two plus three D. T limits 12 X. So the formula for this, if we want to differentiate it because we want to find the interrogative or producing the calculus. So a formula says that differentiation of integration all X F D T T. And the limits are X one to X two. Yeah, this gives us effects to dx divided uh extra minus fx one, the X one by two years. So this will actually give us the value fall differentiation of or derivative of G. S. We can say it or we can write it as G dash X. You think so? Okay, Okay. So we will just get here. We know that here. We will just put healthy function is of ft right function of three. So instead of tv will put extra extra extra small. That is an upper limits we will just put here. So we will get one by X q plus three D by D X of X. That is upper limit rate minus one by global LTD. Instead of in this function, we need to put the global limits so we will get 12 plus three years now. The differentiation of the lower limit lower limit is one, so the differentiation of global limit is actually the constant, so it will be zero here and we will just write it down and for differentiation of X is one only. So the final answer becomes G dash X. That is generating growth of function D. It is one bite execute plus three. This is a ransom.

In this problem we are given with the function G. S equals to integration of one party two plus three D. T limits 12 X. So the formula for this, if we want to differentiate it because we want to find it elevated for producing the calculus. So a formula says that differentiation of integration all X f D T T. And the limits are X one to X two. Yeah, this gives us fx two D X divided uh extra minus fx one, the X one by two years. So this will actually give us the value fall differentiation of or derivative of G. S. We can say it or we can write it as G dash X. You think so? Okay. Okay. So we will just get here. We know that here. We will just put healthy function is of ft right function of three. So instead of tv will put extra extra extra small. That is an upper limits we will just put here. So we will get one by X Q plus three D by D X of X. That is upper limit rate minus one by global LTD. Instead of in this function, we need to put the global limits so we will get 12 plus three years now. The differentiation of the lower limit lower limit is one, so the differentiation of global limit is actually the constant, so it will be zero here and we will just write it down and for differentiation of X is one only. So the final answer becomes G dash X. That is generating growth of function D. It is one bite execute plus three. This is a ransom.


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