why-is-x53-a-correct-antiderivative-of-3x52-whereas-2-x53-is-not-a-correct-antiderivative-of-32-x52

Name: why is x53 a correct antiderivative of 3x52 whereas 2 x53 is not a correct antiderivative of 32 x52
Uploaded: 2019-09-23
Duration: 1 min 39 s
Channel: Numerade
Description: Why is $(x+5)^{3}$ a correct antiderivative of $3(x+5)^{2}$, whereas $(2 x+5)^{3}$ is not a correct antiderivative of $3(2 x+5)^{2} ?$

Question

Why is $(x+5)^{3}$ a correct antiderivative of $3(x+5)^{2}$, whereas $(2 x+5)^{3}$ is not a correct antiderivative of $3(2 x+5)^{2} ?$

Doruk Isik · Accepted Answer

this problem Where us To determine whether he, uh Tiger Ruto Bunch of the Siberian tiger is explosive. Three over X minus 22 plus a constancy. So let&#x27;s take the dare to hope to function on the rifles. Um, Percy have three times x plus three or X minus two spray. Now they&#x27;d ever drop the inter function where we are going to use a potion Group forgot. So the dirigible, the function and numerator is one times defection denominator minus There were there were talks function the denominator which is one multiplied by fortune in the numerator explosive story. You wanna buy function in the denominator square Now we can work. This one s three times X plus three spread divided by explains to spring times. Um, here we have Let&#x27;s see what do we have? A new memory We have explain. It&#x27;s too minus takes My streets of eggs would cancel out and then we&#x27;ll have negative fire. So then we can read This one has negative black. Divine about explains to scrape. Yeah, let&#x27;s combine those terms. Family will have a negative 50 x plus the scribe your X minus two or power And if you compare this with what is given insight into ago. We see that these two are the same so that this is right.

Bobby Barnes · Answer

We want to determine if this equation is true or not sexually. Let&#x27;s put a question mark right here because we don&#x27;t know yet if it is true. But one way we go about solving this, cause we don&#x27;t really know how to tackle this integral directly. Yet we can go ahead and take the derivative of the right hand side and see if we get what is on the inside over in general. So it&#x27;s good and do that so empty by the X bo X plus three over X minus two. Cute plus C. Now, to take the derivative of that first term, we&#x27;re going to need to use first power rule and then change rule. And then it looks like quotient Roy as well. And the derivative or constant shit zero. So we don&#x27;t need to worry about that. See, So first is gonna be three x plus three over X minus two squared and then chain rules to sink the derivative of this. But that&#x27;s gonna be coalition rule. So question rule is low. D Hi. Minus hi de lo All over the square of what is below So x minus two squared and again we just had our constant plus C. Let&#x27;s go ahead and clean this algebra up a little bit. So it&#x27;s three times X plus three over X minus two squared. And then, well, these ones we can actually just get rid of because multiplied by wanted to change anything and then x minus x of these X&#x27;s cancel out and then negative, too minus negative. Three gives us negative five over X minus two squared, and we could go ahead and multiply. The three and five says, Come give us negative. 15. So let&#x27;s distribute that square so it&#x27;s gonna be X plus three squared and the numerator and noticing the dominator. Here we have X minus two. And when we distribute the square, we have another X prize too. So that&#x27;s going to become X minus two. Raise to the fourth. And this is exactly what we have on the inside of our integral. So this is going to be a true equation so we can go ahead and get rid of that question mark because they are equal to each other

Hayden Woerner · Answer

the error in this use of the power rule is pretty drastic. So what it looks like is it looks like this, um, person just took the anti derivative of the denominator instead of actually taking into account that X squared on the bottom is actually x to the negative too. Um, so when you take the derivative of that, um, our I&#x27;m sorry, the integral of that are integral would be, um, five x to the negative one. And then we divide by the negative one and then plus C. So the correct answer would be negative five over x plus c. So not taking into account the fact that that X squared is a negative exponents.

Katelyn Chen · Answer

castle. You can omit the constant when you&#x27;re evaluating definite into girls, and this is because the constants will cancel each other out. You will have after being plus see minus us of a plus C. And so then the seas, and we know that see my C is equal to zero. So that&#x27;s Wat or not, included, not included in their there too much into girl.

Nick Johnson · Answer

we have the integral from 1 to 5 of wealthy X over X minus street. So, um, are function here? We&#x27;re f of x equals while one over X minus three right is not continuous on the close interval from 1 to 5, right. We have an infinite dis continuity here when X equals street when X equals three other nominator here is zero. So therefore this given integral is improper, So improper. Okay, um and then for B, I think we have the integral from 1 to 5, um, of dx over X plus three. Okay, so here be into grand. Well, the function f X equals one over X plus three is continuous again on the close interval from 1 to 5. Right here. Makes the denominator zero is one negative three, right. The negative three is not on. Is not in part of this interval at the interval that we are integrating over. So therefore, the integral is not improper. Okay, um, and then for C So you say this is we&#x27;re not improper, or I guess, proper. Okay, um, for see, we have the integral from 0 to 1 of the natural log of X, the X They shall hear the Integrion while I was having a function. F backs is equal to NFL. Lagerback&#x27;s right. This is not to find for X is equal to zero. So therefore, the function F backs is equal to the next. Lagerback&#x27;s is not continuous on the close interval from 0 to 1. But we haven&#x27;t infinite this continuity at X equals zero. So therefore this given into grow is m proper. Okay, Um, and then for D, we have well, the integral from one to, well, positive infinity of e to the negative x dx. Okay, well, here is kind of I guess karate is Maybe because we have, um um the limits here of integration, we have an upper limit. Um, which is going to infinity. So therefore we right away say, just given integral is improper. Okay, Um and then for party, we have what we have the integral Well, from negative infinity to positive infinity of DX over the cube root of X minus one. Well, here became obvious, right? The limits of integration here right are infinite, right? So we have two infinities impairing in our grand, so therefore, um and, um and also while the in a grand also has an infant. Infinite discontinuity at axis Equal toe one. So, yes, this in a girl is definitely, um, host. Definitely improper. Okay. And, um, lastly, we have part off. So for F, we have the integral going from 0 to 2 pi over four of tangent of acts the ex. Because you hear the in a grand, um tandem X right, um, is well, we are continuous and zipped higher. Afford. There&#x27;s nothing. There&#x27;s no bad. You plug in any tangent of anywhere between on the close interval, going from 0 to 2. Pi over four, right to function Effort backs is equal Tangent backs is continuous on this interval. So therefore this given integral, is not improper. So we can go ahead and say this is a prepper. All right, taker.

Problem

Is $\frac{1}{(x+5)^{3}}$ a correct antiderivative…

Problem 41 Easy Difficulty

Why is $(x+5)^{3}$ a correct antiderivative of $3(x+5)^{2}$, whereas $(2 x+5)^{3}$ is not a correct antiderivative of $3(2 x+5)^{2} ?$

Answer

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus