rewrite-the-given-integrals-so-that-they-fit-the-form-int-un-d-u-and-identify-u-n-and-d-u-int-frac-4

Question

Rewrite the given integrals so that they fit the form $\int u^{n} d u,$ and identify $u, n,$ and $d u$.
$$\int \frac{	an ^{3} x d x}{\cos ^{2} x}$$

Tyler Moulton · Accepted Answer

uh huh. We want to evaluate the following the integral of the tangent cubed of X. Dx over the coastline. Square X. Dx. This question is testing our knowledge of methods of integration in particular. I recently learned method called the power rule. Remember that the power rule for integration states that the integral of U. To the N. D. U. Is U. To the N plus one over N plus one plus the see the constant of integration. So if we identify you do you and end in our given integral, we can easily solve the the power rule. So we see that you as 10 cubed X. Do you as you can&#x27;t square X. Dx. Which is equivalent to our given integral dx over coast squared X. And N is equal to three. So you was actually tan X. So then by the power rule you must have that are integral of U. D. N. D. U. Is U. D. M plus one for over four plus C. Or 1 4th times a tangent fourth x plus seen constant of integration.

Geena Pullo · Answer

thigh. So in this question, we&#x27;re going to be solving our indefinite integral using the method of you substitution. So we have the integral of e to the tan of X times thesis EQ and squared x d x. And so first thing we&#x27;re going to do is we&#x27;re gonna set Are you equal to the tan of X? And now we&#x27;re gonna just take the derivative of both sides. So we&#x27;re going to have d. U is equal to the tan of, uh, start at the Chan, vexed the seek and squared of X dx. And now we&#x27;re going to divide both sides by the secret squared off X. We&#x27;re gonna have d&#x27;you over the seek and squared acts is equal to D X. And from here we&#x27;re just going to replace r 10 of acts with Are you and r d X with R D you over seek and squared of acts. So now we&#x27;re going to have the equation that e to the U times, the seeking squared of x times d you over the seek and squared of X. So our sequins are just going to cross out and we&#x27;re going to be left with the integral of e to the U. D you and e to the u It do you is just equal to e to the U plus c Ah, Now we&#x27;re just going to substitute in for Are you and are you is equal to the tan of X So we&#x27;re just going toe have e to the tan of x plus, see? And that is our final answer.

Julian Gerber · Answer

So here we have the integral of e to the tangent of X times See? Can&#x27;t squared of X Andi, solve this. We&#x27;re going to use a U substitution where we set you equal to tangent of X So do you equals the derivative of tangent which is C can&#x27;t squared axe d x and we have a c can&#x27;t squared X dx right here Weaken Just substitute that for D U or Substitute. Do you infer that? So all we have is the integral of e to the U. D you the integral of either the u is just e to the u plus the constant of integration. And now, just to get the final answer, we just substitute in tangent of X back in for you So we get e to the tangent of X plus the constant of integration

Geena Pullo · Answer

I. So, in this question, we&#x27;re going to be solving the indefinite integral using the method of the substitution. So first thing we&#x27;re going to do is we&#x27;re going to set are tan of the arc Tanne. So I should say X is equal to you. Ongoing take derivative of both sides derivative of the ark 10 of acts is just equal to one over x squared, plus one d x. So now we&#x27;re going to plug in tow our equation. So we&#x27;re gonna plug in Ah, for the tan of, uh, for the ark 10. I&#x27;m sorry of acts, are you? And then we&#x27;re going to plug in for our ah, one over x squared plus one d x r d you So we&#x27;re just gonna have the integral of e to the U D you, which is just going to be equal to the Inter to Ah, you plus C on from here. All we have to do is plug in for are you and are you is equal to the arc Tanne? Uh, axe. I&#x27;m sorry. This is we have e to the u is what the, um integral of each of the U is equal to And so from here we&#x27;re going tohave e to the Ark tan of X plus C on That is our final answer.

Julian Gerber · Answer

here we have the integral of e to the inverse tangent of X over one plus X squared DX. So to do this, we&#x27;re just gonna use a u substitution. So we set you equal to the inverse tangent of acts than do you is just equal to one over one plus x squared DX because the derivative of inverse tangent is just one over one plus X square. And luckily enough, we have 1/1 plus x squared DX. That&#x27;s just this in the denominator here. So we can just change this to e to the U D you now the integral of ET you do you is just eat of the U plus c So all we have to do is substitute in inverse tangent of X back in for you to get our final answer. So yet e to the inverse tangent of acts plus the constant of integration

Linh Vu · Answer

in the But I am The question were given into ground, of course side to thanks to the X in addition uh, superstition. You quit you, don&#x27;t you x Then you go to the Jodi acts and here we can be able to write disintegrate in terms of the new now. So just now we we did you and the exit coaches did you are what you Therefore we have the Akosombo view and then you on with you The Jews, a constant coming in our side don&#x27;t have integrated. Of course I&#x27;m the new Do you the entirely riveted the course I A year If we go to the Cy on your prissy because you go to two X So we get it. Could you half sigh under your express See? And for the back they were given the integral on the I second square under two x on the x square The ex again using the same petition you go to the X Square began to do you a credit your extra x And here we see that we can write it dx in terms under Dean, you on what you ax and now doesn&#x27;t beat you. Therefore we can write this into court in terms of the year now. Ex status, same second square and the new and now attempts. Did you over to X. We see we can cancer. The X with the X and the number juice A constant can be in our side, so we have a good into grow one half here. Second square you d new until the river. There are Mexican square culture that tension X that continue here plus C but because you could your X square. So get a good half tension off that under X square. Let&#x27;s see.

Problem

Rewrite the given integrals so that they fit the …

Problem 30 Easy Difficulty

Rewrite the given integrals so that they fit the form $\int u^{n} d u,$ and identify $u, n,$ and $d u$.
$$\int \frac{\tan ^{3} x d x}{\cos ^{2} x}$$

Answer

$u=\tan x, d u=\sec ^{2} x d x$ and $n=3$

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Basic Technical Mathematics with Calculus

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$u=\tan x, d u=\sec ^{2} x d x$ and $n=3$

Basic Technical Mathematics with Calculus

Heather Z.

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

Heather Z.

Kristen K.

Samuel H.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus