evaluate-the-integral-given-that-int_0infty-e-x2-d-xfrac12-sqrtpi-int_0infty-x2-e-x2-d-x

Question

Evaluate the integral, given that
$\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$
$\int_{0}^{\infty} x^{2} e^{-x^{2}} d x$

Fuzail Shakir · Accepted Answer

Okay, so we use you Devi on VD you to value with this. We end up with X times e to the power of negative X squared divided. But to evaluate this from zero up to in any way plus the integral from zero to infinity off 1/2 eaten the power of negative X squared X Evaluating this one&#x27;s yours if any reason. Enough with zero plus 1/2 time leading to go from zero to infinity and eat the power of negative X X squared. Yes, Thanks. Squared yes to this is equal to 1/2 times of 1/2 times square Too high, which is the square root of pi over four. And, uh, yeah, so were given. We got that this legal disqualified because we&#x27;re given that the integral from zero to infinity and eat the powerful negative X squared accident 1/2 times the square root of pi

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the uncertain question number 33 compared to 5.8. So we asked to evaluate the integral between zero infinity off eat minus X squared over to the ex. So for 32 makes institution well that why equal Thio X over to which makes the X equal to three to Dubai. Tha&#x27;s integral, therefore is equal to you. Well, he&#x27;s the minus X squared over two companies of minus y squared The ex becomes moved to Why the limits? So an ex sequences every wise also a good director on infinity. What effect sequence? Infinity limits express infinity. Then why also goes to Infinity s? This is equal to its groups to most part by the integral That&#x27;s given in the question which we know is half times group I grew to over two times one over route to so right. This is a group of 1/2 Hi

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the uncertain question about 35 chapter five point Hate to ask to evaluate the integral between zero and affinity are screwed X times by E to the minus x the X So, firstly, let&#x27;s make a substitution and let why be equal to the square of X therefore you Why the ex that&#x27;s gonna be equal to the square of earth Ex differentiated 1/2 times, one over the square. Therefore, yes, Gonna be equal to you two quarterbacks d y What&#x27;s going on, Max is just why so as to why do you want this? Integral But with the limits is just gonna be the same because you zeroed in on limits. X goes to infinity, then the square of Axl And then we have why, by e to the minus. Why squares time to the X, which is to why, Why? So we can put the d y either for the to our friend on make that squared. Now we can use integration by parts. So for this was supposed rights out in a slightly different way. Well, you&#x27;re right. It is why times by function, Why e to the minus. Why Squares. Why so interesting? This by parts Leave the why where it is that integrate the y e to minus y squared and that&#x27;s gonna be minus a heart e to the minus y squared, evaluated between zero and infinity minus the interval between their own affinity off the derivative of why would she just want times by the integral of why I eat the finest y squared that&#x27;s mine is 1/2 which we cannot put here to make plus a heart e minus y squared you are that&#x27;s two times by Okay, this square bracket in the limit is why goes to infinity, then eat the minus y squared goes to zero faster than why goes to infinity Suster script zero than minus evaluated at zero eat the zero, which is one. So the y part it&#x27;s gonna make that zero plus 1/2 And then this integral here is justly in school that were given as a hint in the question. So that evaluates square e pi over too, huh? So, of course, this half and this too we&#x27;re gonna cancel zeros could vanish and we&#x27;re left with square pie divided by two

Linh Vu · Answer

because the argument in the covers from 0 to 1 needs you. The X minus one over each of that you x, the x In this question I&#x27;m going to use this institution. Where would you go to each of the X and then you go to any to the X the X We can register in June. Do you d x on? So when ex ego Gisele, we get a year, you go do one when ex a good Jew one and then we get in. There you go, Juke e And from here we can write disintegrate into the form from one to e. And then we have the new minus one over year square the accidentally Could you do you all over you? And then we get doesn t go to integrate from one to e and we get the humor minus two minus on uber minus three Do you? And then we get the entirely rift of the 1st 1 Nico Judah, One month is one over you the second contender plus one on the Do you square Now he violated and the one to a so it would with a inside again. Uh ah. 11 over to a square, minus one of a and then my Enough. There we have 1/2 minus one. Then we should get this when we co two Ah, we have him. Will mean that to a square. We had gotten one minus on two e and then we have him would be, ah, plus 1/2. And that&#x27;s when we

Fuzail Shakir · Answer

Okay, so the first thing we&#x27;re gonna do is we&#x27;re gonna do integration by parts. When you do that, we end up with negative X squared plus one e to the power of native acts, plus two times the integral of X E to the power of native X T X. So this means that we have to do the integration by parts again. Once we do that, we end up with negative X squared plus one e to the power of native acts, plus two times negative x e to the power of negative acts, plus the integral of e to the power of negative X t X. So once we do that and then simplify the sound, we end up with Native X Square minus two X minus three. The whole thing is multiplied by C to the power of native acts. And now we have to evaluate all of this from 2 3 0 to 1. If we do that and we plug in numbers, we end up with negative six divided by E plus three. So our answer is three minus six already

Robert Daugherty · Answer

Section six. That one problem number 18. We have a definite integral and we need to use integration by parts in order to make this work. One of the first things you can do when I look at this and say Let&#x27;s split this up into two different integral So 01 x squared e to the minus sex D X plus the integral from 0 to 1 e to the minus x dx. So immediately see this one? Okay, that&#x27;s trivial. I can do that one this first integral right here. This is the one that I need to use integration by parts in order to make this work. So let&#x27;s focus on the indefinite Integral. How would you integrate X squared e to the minus X de eggs while using integration reports You would let u equal x square The polynomial term do you is going to be two x t x devi is going to be e to the minus x t x. You integrate that to get minus e to the minus x so that tells me that I can rewrite this as X squared e to the minus. X t X is going to be equal to youth e so minus X squared e to the minus X minus The integral VT you So that&#x27;s going to plus two temptation agro x you to the minus X DX. Now we need to do the same thing again. Integration by parts you is equal to X. Do you The good to DX D V is equal to minus eat of the minus X t X. Therefore, V is equal to minus E to the minus X, so that tells me that this integral becomes minus X squared e to the minus X plus two. And now we do it again. You v minus integral VT you So that&#x27;s minus X E to the minus X and then minus the inner grove et you is just plus the integral of e to the minus X dx. And so this becomes minus X squared e to the minus. X minus two x e to the minus X minus e to the minus X plus a constant of integration. So now I&#x27;ve done integration by parts to, um integrate this Ah, 1st 1st integral Um, and when little mistake here is when I did this integration, this too was distributed. So this is gonna be minus two. So let me correct that. Minus two. So that is our integration. So now let&#x27;s go back to our original formula. So the original formula we were looking to integrate the integral from 0 to 1 x squared plus one e to the minus x dx. We&#x27;re broke that into two separate inter grows 01 x squared e to the minus X dx plus integral from +01 e to the minus X dx. Now what we have just figure it out is we figured out this in a grand by doing this integration. So what that gives me is that the integral from 0 to 1 x squared plus one e to the minus x dx is going to be equal to So this is minus X squared e to the minus X minus two x e to the minus X minus two e to the money&#x27;s X And then when you integrate this last integral, that&#x27;s gonna be minus E to the minus X that all of this gets evaluated from zero toe one. And before I do that evaluation, there is some simplification that can happen. So we can factor out a minus e to the minus X And so this just becomes X squared. Plus two X and then you see, that&#x27;s minus three Eat of the minus X of factory at the minus sign. So you get this so this gets evaluated from 0 to 1. So now it&#x27;s just a matter of doing arithmetic and not making any mistakes as we go along here. So let&#x27;s just substitute a one and a zero when I substitute a one minus E to the minus one. So that&#x27;s minus one over E and we use substitute a one into that polynomial you&#x27;re going to get one plus two plus three and then when we substitute zero e to the zero is going to be one s. So you&#x27;re going to get a minus one. And when you substitute a zero, you&#x27;re going to get a three right there. And so this becomes minus six over E plus three. So the answer is three months six over E, which is the answer for that definite integral. So again, the key here Waas Looking at this, you had a polynomial times an exponential. It was very easy to split this into two differentials. Get to in a rolls. This interval is easy to evaluate. This Inderal requires integration by parts twice because it&#x27;s a second degree polynomial. So then it was just doing the calculus to do that integration coming up with an answer and then the arithmetic. Try not to make a mistake to get the final answer for the definite integral.

Problem

Evaluate the integral, given that $\int_{0}^{\in…

Problem 34 Medium Difficulty

Evaluate the integral, given that
$\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}$
$\int_{0}^{\infty} x^{2} e^{-x^{2}} d x$

Answer

$\frac{\sqrt{\pi}}{4}$

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James Stewart

Biocalculus Calculus for the Life Sciences

Grace H.

Catherine R.

Caleb E.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

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$\frac{\sqrt{\pi}}{4}$

Biocalculus Calculus for the Life Sciences

Grace H.

Catherine R.

Caleb E.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

James StewartBiocalculus Calculus for the Life Sciences

Grace H.

Catherine R.

Caleb E.

Samuel H.

James Stewart

Biocalculus Calculus for the Life Sciences