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(sin(1))2+1 dx r_10 points) estimate the integral...

Question

(sin(1))2+1 dx r_10 points) estimate the integral

(sin(1))2+1 dx r_1 0 points) estimate the integral



Answers

compute the definite integral. $$
\int_{0}^{1} \tan ^{-1} x d x
$$

And this problem, we're solving the integral of DP Divide by four minus rupee from 0 to 1. So again we're gonna use a substitution You equal Teoh route. Actually, let's go ahead and take the whole denominator. So let's do you equal to four minus rupee and do you is gonna equal 1/2 rupee, actually, make that negative dp we saw for Rupee appear Get through P is equal to four minus you. So we'll combine these two to get negative actually will distribute. That's will have two U minus four do you is equal to TP So let's look at the limits real quick. At zero, we're gonna get four minus zero, which is four. And at one we're gonna get four minus one, which is three. So on top, we have to you minus four. Do you on the denominator? We just have to You will distribute this one over you to both terms and evaluate. That's good. And while we're at it will switch the women's of integration. So negative. Negative. And what? Flip it So 34 negative out front. See, we have part two years. Well, Howard said that so too minus eight over you, do you? We'll evaluate that. So let's go ahead and distribute the negatives. Negative. Positive. So negative to you. Plus a Ln of you on 3 to 4. This will be negative too. Fan will evaluate for the a l n of you so plus eight Ellen for minus eight elling of three. We can combine this they get up to plus eight Ellen of 4/3.

Okay so we're going to be solving this definite integral and we're going to use integration by parts to um do it. So here we're going to let you equal X. And T. V. It's going to equal three to the X. And we'll see why we let u equal X. Um and DV equal 32 X in a second. Um So if you is equal to X then D. U. Is going to be equal to one and F D V is equal to three to the power of X. Then V is going to be equal to 3 to the x divided by the natural log of three. And so now we can see that um if we use the formula to find the integration by parts, guess I'll first move these up so I have some room. Then this integral becomes U. Times V. So X times three to the X divided by the natural log three and then minus the integral of the time's D. You so that would be three to the X Divided by the natural log of three, the X. And so now this becomes a doable integral. And we can see that the reason that we let U equal X here is because then D. U. Is equal to one which makes this integral um a lot easier to do so. And now that we have it in this forum, let's go ahead and do this integral. So you can say that this is equal to X Times 3 to the x divided by The natural log of three minus. And then this is going to be 3 to the x divided by The natural log of three squared since we started this natural log of three. And then the integral. Three to the X. Is three. D. X divided by The Natural Log of three. Um And now we're looking on the interval from 0 to 1. So when this is equal to one we're gonna get three divided by the natural log of three minus three divided by the natural log of three squared. And then when X is equal to zero, this term is equal to zero and this term goes to negative one divided by the natural log of three squared and that's going to be minus. So it becomes plus one divided by The natural log of three squared. And so We can simplify this to three, divided by the natural log of three -2, divided by the natural log three square.

You want to calculate this iterated integral. So first we're goingto enter it with respects to X. But what we're going to do before we do that is to use substitution. This is isn't necessary. But a minute anyway, just to clear up any cash in there are there is. So from zero to one, our ballons, we're going to change accordingly. We're changing our integration variable. Why eat you? Do you do all right? Took using a fundamental fear of calculus. This e comes how, uh, becomes this. Were re evaluate from negative. Why? Toe to minus y all right. And, uh so we've got we actually end up with two into girls, and I'm going to write them. Split up. Has to intervene. So, Teo, do this portion now. So we did it. Use up for this. We're gonna have to do an integration by parts not only going to write one of the integration by parts down because they're very, very similar. So new is going to be equal toe. Why? Devi is going to be equal to e to the negative or each of the two minus y. Why do you is equal to you? Why and Devi is equal connected. E to the to minus one. See? All right. And very similar for this, isn't you? So what we end up with after we do our integration by parts is negative. Why e to the negative too? Why are you eating the negative? Each of the two minus y minus the integral from zero to one. I leading this guy also from zero to one. Uh, negative e to the to minus y. Do you know why? That's his first integration by parts minus when we do integration by party. So very, very similar. We're just not gonna have this to here, but the negative. Why he is the negative. Why I waited from zero of the one minus thie, integral from zero to one of negative deeds is a native. Why? Do you know why? Okay. And, uh, this after we wait, do these inter girls here and here. These ones underlined in green. You can do those intervals pretty easily. So we're gonna get native y E to the negative or into the to minus y minus he to the to minus Warner into the two minus. Why? Plus why e negative wide plus, even with the negative, why we're gonna evaluate this entire thing. Zero skip a few out of her steps. Hopefully see. What way did here we have this negative. We distributed weight distributed this negative at this. Any of this negative or negative interest? This is another negative. So it's a posse. Did it. So we've got this here, and what we want do is we want you No, evaluate this. So, uh, let's rewrite this little the first two terms. We can take out negative e to the negative, too. Why? Just with y plus one and the second portion, we can factor out e to the negative. Why? And we're left with y plus one. And we're value ants from zero to one. And in fact, if we want, we could do a little more. Why, that's one e to the negative. Why? Why is he to the children's ward and evaluating this from zero to one? So when we evaluated at one, we're going to get to time to eat the Knight of one ninety two one. And then when we evaluated at zero today, we're gonna get one times e to the negative zero. Your mind is your own right. So this ends up being two times one over E minus, C minus one minus square. Some believe the answer like that. We could do some more. A simplification, but there isn't actually.

1.7 Next to the 1/3 DX and I were looking at the integral from 01 We know we can divine are you? And then we can define our d acts. VX is do you over seven. Given this, we can now write or integral in terms of you. And now the upper limit is going to be you is one plus sometimes one, which is eight. So we now have the integral from 1 to 8. Okay, let's pull up the constant, which is 1/7. Yeah, and now we know that we can use the power rule which is increased the expert it by one and divide by the new exponents. And now we're climb a fundamental theme of calculus by plugging end our bounds to get 45 over 28.


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