integrate-each-of-the-given-functions-int-2-e-4-x-d-x-3

Question

Integrate each of the given functions.
$$\int 2 e^{-4 x} d x$$

Tyler Moulton · Accepted Answer

Oh we want to integrate. The following. The indefinite integral of two equals negative four X. Dx. This question is testing our knowledge of methods of integration. We&#x27;ve already learned about the power rule for integration as well as the log arrhythmic integration in this section we&#x27;re being evaluated on our newfound knowledge of exponential integration which states that the integral of either you D. You is either you plus C. So we can plug into the solution formula Either you plus see if we simply figure out what you do you are in our integral. So since you is whatever is raised the exponential power it must be. There are integral is new equals negative four X. D. U. Equals negative for the X. We see that we&#x27;re missing a factor of negative one half in our original integral which must carry over to our final solution. Thus are integral is equal to negative one half integral to the U. D. U. Or negative one half is negative four X. Plus the constant of integration. See

Ramzi Deek · Answer

got integral of X squared times Eat the four x dx So we&#x27;re gonna use X squared as you and each the four x dx ez devi So you is X squared and do you is two x So Devi is e to the forex. In that case, V is e to the four x over four. All right, now let&#x27;s we know that integrate by parts, it&#x27;s integral. A few d v is equal to ah you v minus. And to grow off v do you? So let&#x27;s start, uh, putting in what we have so x squared times e to the four x over four times integral of v do you? So that&#x27;s two x times e to the forex over four. So let&#x27;s cancel out two and four. That becomes too. So we&#x27;ll take half out. So x squared you to the forex Over four minus God. Put the minus here minus and to go minus half integral of x E to the forex. Well, now we still have X times eat forex. So again, we&#x27;re gonna do this integral by parts. So again, I&#x27;ll take you as X and Devi as e to the forex. So Let&#x27;s ah, derive you. So do use one. Now that we have it as one, it will be easier for us to salt from there. So do you use one V will be Eat the four X over four. So let&#x27;s move from there. So x e to the forex. Uh, over four minus half into Remember, it&#x27;s UV minus view. Do you? So you is X each the forex over four minus into go of V, do you? So that&#x27;s also eat to the four X over for so x e to the forex over four. Remember, this is what we have here. Keep it a squared two X squared e to the forex. Over four, minus half ex. Eat the forex over four minus. So integral feet. The forex over four is each of forex of a force that becomes e to the forex Over 16. Uh, you can distribute half. Uh, that won&#x27;t be no sorry. Though you might know you can keep the equation as such. Her So add your constant here at the end. So this is your final answer.

Sid Wan · Answer

Okay. Finds an integral off That&#x27;s over full. Tend Actual full square the ah, that&#x27;s C If we take at our four square you to be that he was Explorer over sixteen D&#x27;You equals axe over Ate the X so we can run this a cz This guy is to see you, Tad. You too. Do you on your hands? You Because ah, well two in front collective national off cause I you we&#x27;LL see if you plugging you to be x square over sixteens that give to natural log because I asked Square over sixteen c, yeah.

Aishwarya Krishnakumar · Answer

so for problem 11. We are asked to use substitution to find this indefinite integral. So for the U substitution, we&#x27;re gonna let this equal, are you? And we do that because it&#x27;s a function inside of a function. This function is inside that e function. So we&#x27;re gonna let you equal to X to be the fourth power. So we go ahead and take the derivative of both sides. We get to you equals two times for ext three d x. So to make this easier, we could do do you over to divide both sides of the incursion by two. He goes for X cube, deed X. Now we go ahead and we&#x27;ll get in like us. So the four first weaken do this part and it would be four x cubed de ETS. But we have ah 1/2. So we remove that outside the integral and all of that equals you. And then for the you part, you would do e to the you power because that is what we chose there you to be This equals won her time&#x27;s a So the integral of this e to the youth is even to you and then you have to do the plus c. We&#x27;ll be eating you over to you plus C and you can&#x27;t leave it in terms of you. You have to plug in two X to the fourth in replacement of every U to exit before two times two weeks before we&#x27;ll see. And if you simplify this, you get to e to it to the fourth over for it&#x27;s to the fourth. What? See? And this would be your answer.

Robert Daugherty · Answer

7.1 Problem number 16 Here we&#x27;re dealing with integral is and basic techniques for integration. In this case, you look at the Inter Grand and it could be simple or if we had fewer termed with the E to the X. So the key is gonna work on the one that has a negative sign there. And yes, you can You can look at this a z to the two x over e to the two eggs minus four e to the minus x multiply everything by E to the X over e to the X and then that transformed this integral. What you see on top will be e to the three X over each of the three eggs minus four DX. Now comes the time for you to change the variable integration, so let u equal e to the three X minus four Indy. You is equal to three e to the three eggs D x. So if I look at this integral, I could have sort of have each of the three x t x over e to the three x minus four. If I saw three e to the three acts, then that would put this in the form that I need. So now what? You see, this is just the integral of one over you, d you and that&#x27;s a quite easy one. That&#x27;s the natural log. So this is 1/3 natural log of you, plus a constant of integration and then go back to your variable for you. So this is 1/3 natural log eat of the three X minus four plus constant of integration.

Amy Jiang · Answer

Okay, So it&#x27;s notice here for in ago that we have hyperbolic hotel agents. Aunt Hyperbolic. Does he get so notice? Or recall that the derivative of hyperbolic contingents that&#x27;s equal to negative coast chickens. Spirit of X. So I don&#x27;t leave the right Or why don&#x27;t we use you? Stop. We cannot you be equal to the following and then do you is equal to negative? Have got a hybrid ball coast against. We can bring that to the other side to negative. Yes, you can squared X for this portion right here. This is equal to do you and then we have intentions squared of you are acceptable to use word so he gets the integral of you squared. Do you taking the interval? We get YouTube over three. But see, when we can get you in terms of X, that is hyperbolic. Coach engines cubed, abets over three busting

Problem

Integrate each of the given functions. $$\int_{-2…

Problem 6 Easy Difficulty

Integrate each of the given functions.
$$\int 2 e^{-4 x} d x$$

Answer

$-\frac{1}{2} e^{-4 x}+C$

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Caleb E.

Kristen K.

Samuel H.

Michael J.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 28

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Caleb E.

Kristen K.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$-\frac{1}{2} e^{-4 x}+C$

Basic Technical Mathematics with Calculus

Caleb E.

Kristen K.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Caleb E.

Kristen K.

Samuel H.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus