integrate-each-of-the-given-functions-int_0pi-4-frac2-sec-2-x-d-x4tan-x

Question

Integrate each of the given functions.
$$\int_{0}^{\pi / 4} \frac{2 \sec ^{2} x d x}{4+	an x}$$

Vikash Ranjan · Accepted Answer

In the problem we have to find the integration And the limits are zero and by upon four. So it is twice sex squad X. The eggs. You wouldn&#x27;t buy four plus 10 eggs. So here let us assume four plus 10 X equals Q. Or it is sex square X. D. X equal D. Q. Now this is equal to twice integration. Food and five did you upon Q. This is equal to twice. Ellen Mark Q. four and 5. There 40 days twice. Ln five minus twice. Ln four. So it is twice Ln five upon four. So this is the answer to the problem.

Vinnu M · Answer

So in this problem we simplified the giving it equal. To do that, we take a variable U where u is equal to panics. So we have do you as he can&#x27;t square x gxe so the original and eagle will become Peter toe one fruit off you, do you So after solving separate. So after solving, we have you raised toe three by two. But the limits up from 0 to 1 and after solving get the entire s.

Audrey Fong · Answer

we want to integrate dysfunction a couple of minutes This But for this question, we will use substitution method. Let you he closed with danger. Next. When I differentiate my you, I get Seacon Square X g x See my sequin square and my d excess here because this is definite integral. We need to replace the lower upper limit as well. So when X equals zero My used engine zero. And that would be zero when expertise powerful, my newest engine powerful. And that will be one. So let&#x27;s do the congressman now. This will be zero and this will be one eat about tension, actually Eat about you and Seacon Square X g x will be Do you? So when we integrate either Paul, you get betting self within 10 So we just some Indies. Well, just get he minus one is the final answer

Geena Pullo · Answer

Theo the integral of the seeking of X pie and the tan of pi x dx. So first thing we&#x27;re gonna do is your You substitution were going to say you is equal to high ex on now and take driven of both sides and have d. U is equal to high of D X and over divide both sides by pie and we&#x27;re gonna have d&#x27;you over. Pi is equal to X and from here we&#x27;re going to just substitute this into our equations. We have the integral from 0 to 1/4 of the Sikh and now we&#x27;re gonna have you because you&#x27;re going to subsidy in for our pi X on the tangent of you. And now we&#x27;re gonna sit in our d you over pie for r d X on from here We can take out our one over pi and we don&#x27;t have the integral from 0 to 14 of the Sikh and of you tangent of you D you on from here? We can just take the integral on. This is equal to be second of you from 0 to 1/4 times one over Hi. On dso from here. What? We&#x27;re gonna do is gonna substitute back and for you and are you is equal to hi axe. And so we&#x27;re going to have one over pi times the seek and of pi X from 10 to 1/4. And so now we can substitute in our values. So we&#x27;re going tohave are won over by all of this. Ah, Now we&#x27;re gonna have the seek and, um pi over four minus the sequent of pie time zero, which is zero. And so the seek and of zero is equal to one over the coasts of zero and close zero is equal to 12 This is equal to one, and the secret of hi over four is equal to one over the coast of pirate before which is equal to one over the square root of 2/2, which is equal to serve to sorry two over the square root of two which, when we get rid of our screwed in the denominator, is equal to the square root too. So we&#x27;re going to have hi times the square root of two minus pie because this is equal to one on. This is equal to these very to and that is our answer

Enkhzaya Enkhtaivan · Answer

That&#x27;s his problem for the 648.3 and we have a definite integral that goes from the Pirate for two, power for six times 10 Geant to fourth Ex ex. So that&#x27;s it&#x27;s been Titian, too. They in definite advantage Rito of tension to the fore Thanks as usual, whenever there is an even power of tension our second, it&#x27;s always a good idea to think about that in terms ofthe secret. So what? Where, when I mean by that is we would write it as tangent squared of X times engine squared off x, the ex and we know that here. Tension square axis single second squared of ax France one fine, dear One of their relationship to intention This sequence So this will become this will become this. So the count into off changing squared off axe times, second square T X Rangers internal off Changin Square Thanks d x here And so the 1st 1 The first Angela attention squared of acts times sick and squared off X dix. Well, one thing again so that there&#x27;s another relationship to intention is seconds which is basically sick and squared of X here. So second squared X here This is just the off tangent squared engines off X which essentially just the recent degree student of the fact that tangent threw it over attention Next this second square x so far. But for a 2nd 1 we will use thie thiss formula here began. So this is again second squared of X, my swan. So after applying this to these two will see that the 1st 1 is just attention squared axe and the differential ofthe tension Axe. Now this&#x27;s and this is immediate. We can immediately calculate this, but the second term becomes threw off. Second squared X the eggs plus constance function won t x Right, So this will be count Tangent Cube of X derived by three and minus war again Here, descent getting through here Seconds seconds squared of axe here. Well said which sausage? Remember the fact that this second squared So attention tw attention well again here. So we just gotta remember this fact again, this is this is just do it off tension effects. So this will become just tain geant off acts and plus X plus. See, So now that we know the anti, do it, we can just play in the balance to get are finalized. So considering the second page here Sorry. Interval six times Native Pi over four, two hour four ancient to fourth off x The X single too. Six times seeing tension to bex over three. Attention Cube of X three Honest attention. ACS cross ex divided at the boundaries pilot. For now, they were forgetting the constant here because now we&#x27;re dealing with a definite interests of this system. Six times were thie around. We have we have to need me to know the value of tension of power for which is precisely one. So this is 1/3 nice one class one minus six times the value of tangent of native pine reform is NATO. One is tangent. It&#x27;s about function, right. The finishing so minus again dated one for us NATO. So don&#x27;t fight this. The first turn look stupid. So it&#x27;s going to be just too. Why minus six times this is, um, negative 1/3 because the last two terms cancel each other out here. Here. So it&#x27;s native. Six is screaming for us to okay. Forced to find

Robert Daugherty · Answer

section eight dot to problem number 46. We want to find the definite integral from negative power before to pyro four of six times. Attention, fourth of X d x. So first thing I like to do is just to figure out what this, uh, anti derivative ISS So how would we integrate six times a tangent? Fourth of X d x will. Often one of the keys is to write this in powers of two changes squared of X changes squared of x d x And then I know that the changes squared is seeking squared minus one. So this is six seeking squared of X minus one time&#x27;s a changing of x d x And if I multiply Excuse me. Changes squared. If I multiply this out, this becomes six. Integral seeking squared of X tension squared of X um, and then minus six Tanja squared of X T X. Now again, in this second Inderal, um, Tangent squared is seeking squared of X minus one. So this gives me so six seeking squared of X 10 squared of X and then minus six. And then let&#x27;s just write this a secret squared of X minus one the ex. And so this becomes six sequence quite of X tens. Great of X T X minus six. Seeking squared of D x X t X plus six integral d x. Okay. And so now in this first interval, I can let you equal the changing of X. Then d&#x27;you is equal to the secret squared of X d. X. So this is six times the integral of you square, do you? Minus the integral. The second squared is just the tangent. So this is minus six, um, tangent of X and then plus six x plus a constant of integration. So this becomes six. You cubed over three. And so this becomes, uh, two. You cubed minus six tangent of X plus six six plus C and us equal to the changes of X. So this final answer is to tension cubed of X change acute of X minus six tangent of X plus six x plus a constant of integration. So that is the anti derivative of the six change and forth of X d. X. So that&#x27;s what I&#x27;m gonna need here. So if I go back to my definite integral now, I&#x27;ve got everything that I need to Ah, to make this work. So this is going to be too Tanja. Cute of X minus six. Changing of X plus six x evaluate from negative pi over four to power before. Okay. Yeah. So let&#x27;s go ahead and see if we can make this work. And now it just becomes an arithmetic problem. Okay, so I&#x27;m going to have something like this. So if I substitute pie over for the tangent of Pyro for is equal toe one, So I&#x27;m gonna have a two. And that 1st 1st expression there. And so then, too, and then minus six. So two minus six and the empire or six pi over four is going to be a plus. Three halves pie. And then if I substitute a, um minus pi over four and so I&#x27;m gonna come up with there&#x27;s that minus pyro four. The tangent of minus pi over four is a negative one. So I&#x27;m gonna have a two times a negative one cubed, okay? And then I&#x27;m have minus six times and negative one, and then I&#x27;m have minus, um, six pi over four. So that&#x27;s minus three halves pie. So this becomes three House pi minus four minus. And then what I have here? I&#x27;ve got what? Six minus two. So this is gonna be four minus three has Hi s. So this is going to be, um, go to the next page. This is three halves pie minus four. This changes a plus three halves pie money&#x27;s for plus three halves. Pi minus four. And so this is six halves pi minus eight, which is three pi minus eight. And that is our answer. So three pie Money&#x27;s eight is the answer for this definite integral. So we just write the answer here. So this is three time on his eight and again the key waas. As with many of these, Tange is being able to take that derivative by organized

Problem

Integrate each of the given functions. $$\int \fr…

Problem 12 Easy Difficulty

Integrate each of the given functions.
$$\int_{0}^{\pi / 4} \frac{2 \sec ^{2} x d x}{4+\tan x}$$

Answer

$2 \ln \left(\frac{5}{4}\right)$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Anna Marie V.

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Precalculus Review - Intro

Functions on the Real Line - Overview

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$2 \ln \left(\frac{5}{4}\right)$

Basic Technical Mathematics with Calculus

Grace H.

Catherine R.

Anna Marie V.

Heather Z.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Catherine R.

Anna Marie V.

Heather Z.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus