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Evaluate the double integral = Gf;(# + 15) dydzAnswer:...

Question

Evaluate the double integral = Gf;(# + 15) dydzAnswer:

Evaluate the double integral = Gf;(# + 15) dydz Answer:



Answers

Find the following.
$$\int(15 x \sqrt{x}+2 \sqrt{x}) d x$$

That's the last to find the integral, the full in equation dominos here that we have radicals here and we don't have any other way to find the inter girls of a radical besides turning them into an exponent. So let's start returning that he's too into exponents. So we know that the square roots that's equal to X the power of 1/2. Okay, I noticed here that we have X Times X acquired 1/2 to using our product. Herbal. This is equal to Exeter have one plus 1/2. So that's 2/2 plus 1/2 which is equal to accept card 3/2. So it's ready that So do we get extra power of 3/2. I know it's finally used or, um, are some world. So we get in a world falling. Okay, I know notice here that we have Constance next to our external. So let's use our constant multiple room to bring them outside. So we get the falling. Okay, So now let's you apart. Roll. So we get 15 times X to the power of 3/2 parts. 11 is to over to, and then two times X to the power of 1/2 plus one. That's to over to then we have pussy and I'll it simplifies to get 15 times extra part of 5/2 over for over two plus two tons. Exit a bar of three over to over 3/2 and in policy. Okay, so what's it? Simplify. So we have 15 divided by five over too. Well, that's equal six. So we get six x two par 5/2 plus for over three extra power 3/2 plus c. So this is our situation.

Okay, so we're giving the following integral to evaluate, right? And so let's just play with the denominator a little bit. See if we can get it close to something that we can work. So this is nearly a square. It sort of looks like a bunch of squares. So let's try and make of the square. So what if we write it like that? Although we get four times four X squared plus or X plus one plus one. And so this is four times two X plus one square. This is two squared. So if you rate this is two squared, you can bring it inside. It becomes four x plus two squared course, one watts one. And so, no, you know that this is the interval of four exposed to squared plus one. So that is, Let's see here. This is nearly so let's see if this is the right answer. Let's just guess and check So we know that when we have something squared in the denominator plus one, that's the art tension. However, we've got a changeable going on here, so when we differentiate this, we're going to get four in the top, and then this war thing in the bottom, so we need to divide by four. So the answer is 1/4 are tangent or X plus two and ah.

Okay, so let's find the falling in a girl. So if you try to use use up, we'll notice that in our numerator as well as a denominator, you have a next term. And if you take about to be you, then r D X is just a constant. So we can't really use use up here. But notice here that we have in our numerator a two X plus three so we can break this up into two factions. That is, we have the integral of two X over exports seven as well as, uh, we actually need the X here, plus the integral of three over exports. Seven the X. Okay, so why don't we that, um, you contest out you We can let that equal issues exports summit and then do you that's that's equal to the X. Okay, So that gives us, um oh, you That's for both foreign girls. So we get the integral of two X we can rewrite, Um, you, in terms of that is a is equal to U minus seven. So we have you might seven over, um X plus seven. That's you. And then we have to you and then we also have the integral of three x 47 It's you and then we have to you. Okay, so let's split. Split up this in a row once again. So we get the integral of you over you, which is one to you. And then we have minus in a role of seven over you. Do you? Plus the interval of three over you. Do you Okay. So integral of one is this equal to you? And then we have minus in a role of seven over you. You know that one over you is alone. That's Ellen of the actual out of you. Plus three Elena Veldt Ellen of you. And in Plessy. A knowledge plug in. Are you? We said you is equal to X plus seven. So that's would replace what we have here. Okay, so we get the following solution

Using either in ventricle table or some kind of calculator to find this indefinite interval. Well, we can use table three and six B, and we might notice that Formula fifteen looks pretty similar to what we have. So for me, the fifteen is the integral of the square root of expert plus a squared with respect, X. And this here is equal to this long expression here. So we have our X squared here. And so the other part could be our ace squirts or a squared is supposed to be fifteen. And just looking at this never where we havin a it's always squared. So at least in this case, I won't solve. For a so everywhere there is a squared, I'm goingto plug in fifteen. So this is going to equal X over two square roots of X squared, plus the team wass fifteen over two times the natural log of absolute value of X plus the square root don't X squared, plus fifteen for some constant C. And this here will be our indefinite triple using the table three and appendix


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