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Consider the integral z(r? + )dr:Evaluate the integral by multiplying out the integrand. Evaluate the integral using substitution: Did You get the a answer fOr part...

Question

Consider the integral z(r? + )dr:Evaluate the integral by multiplying out the integrand. Evaluate the integral using substitution: Did You get the a answer fOr parts and (b)? If not_ which one is correct? Can YOn explain the discrepancy?

Consider the integral z(r? + )dr: Evaluate the integral by multiplying out the integrand. Evaluate the integral using substitution: Did You get the a answer fOr parts and (b)? If not_ which one is correct? Can YOn explain the discrepancy?



Answers

Find $\int 4 x\left(x^{2}+1\right) d x$ using two methods: (a) Do the multiplication first, and then antidifferentiate. (b) Use the substitution $w=x^{2}+1$ (c) Explain how the expressions from parts (a) and (b) are different. Are they both correct?

We have the integral of two x plus three squared, and there's actually two ways to solve this problem. The first way, which is probably the easiest, is to apply a use substitution by taking you to be two X plus three. That way do you over to is equal to DX, and we can rewrite this integral as one half times the integral of you squared, which is equal to 1/6 times you cubed, which is, um, which now we replace you by two x plus three and then add C. So that's the first way. The second way to solve this problem would be to multiply this expression out, which is straightforward. If we do that, we get the integral of four X squared plus 12 X Plus nine, which is equal to four thirds X cubed plus six X squared plus nine X plus C. And in both cases we have a cubic polynomial. The only difference is this constant C. So these functions differ by a constant, because when we take the derivative, we get the quantity to X plus three squared, but the constant gets killed off. And so let's write that here. Let's put, um, that one and two only differ. I see, and that completes the problem.

Into question. We're given the integral off the five x minus one square. The X now for the method one we need to expand this square and again. Five X square minus jewels Temp Answer. If I X Times one then plus one square The X they were getting Could you integral under 25 X squared minus 10. Thanks. That's one. The X on Ben the until the riveting in the first term in Coachella. 25 times. Expect tree over three for the second time in getting coaching that five Next square. Infinite last one. Getting put your X plus c and now for the method to instead use the substitution. Knew it could get a five x minus one. Then the new eco to five d x And from here, which again the integral will becomes the new about you. And now the exit coaches do you over five. So we can fucked and we're coming the constant outside and we have your square. And did you? Then we get equal to 1/5 and then and we have the year about 3/3. Let's see. And because you go to this one so we getting good. You won over five. We fit in here. They will have a five x minus one. I am a tree plus c were constancy to this would be the single one So tingly that you answer here they would you for by any constant only so in which would expand it. And then we see only constantly be different.

90 saying that the integral off the dig you over you square time squared under your square plus or minus the square it will echo to the minus are blast And then when the squared off the U squared plus minus a square even in my square times dio Elsa constancy. And in the question where even the integral on 30 extra over the X square Tim squared off the X squared minus two. So we look in the formula receding the number to hear echoed through the square and also where the minus here. So we need your brother from the fund bottom one from my formal away Incontro fraction here squared off the excrement us Jill dividing by down the a square Good unit to X, then present constancy.

From that day when we had into girl off the square root 100 square. When this new square over the new square he knew they were going to go to the minus squared under six grand, minus new square. When you minus ness, I inverse off you are they? That's a constant C and now in the question, were given. Yeah, he's the integral off no squared off four minus X squared all the X crab deep next. So competitive Formula way said that the number four here we go to the square and therefore means that I got to and we can have made a formal at the end of minus square on the four minus X square on the X minus, science offers off the amplitude as a constancy.


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