evaluate-the-integral-int-fracd-xx2-sqrt9-x2-4

Question

Evaluate the integral.
$$
\int \frac{d x}{x^{2} \sqrt{9 x^{2}-4}}
$$

Shamshad Waris · Accepted Answer

we really can do that. Given hitting girl, Be extreme. Identify excess power turned on nine. It&#x27;s a square minus four. Here with this obscure it is gonna go. Who by Kitty Secular ReNu substitute this thing in this battle, then you Then we can remove their spark from so different c A grocer. So the excess value Who by TV and you know very bad people Septic Sepp Tita into Antifa Do check up now I live in the village off eggs and beer in this Integer them Then it will. Yet who by TV Septic in Quetta, Into Vicente. Divided by who? By TV Sector Taka Kula Sweat in tow under nine into Huawei Fealy said Victor in the fullest. Where nine us now simply by this express er then yet then he tough divided by Dubai Kitty Septic Tough from here you will get who then kicked us. Did you No cancel the tank it up from the numerator and denominator. Then you will get Citibank full. I do know one by 60 critical market. So you tend to be I don&#x27;t know the integration of the porchetta sank it so we can&#x27;t a pity. Sanjay Gupta blessing seed An integral concept. Here we are substituting excess tickle toe to buy three. Secretary from here we can I say, if you take a look at the X by two So also we can I work? It&#x27;s logical. Tried to know what is that Both you in Garza Secretary. So we&#x27;re gonna have to pay. Really? But their feet, the menu off the sank in the solar will compute that then loves Ancheta. So we can I sank a tragical route under nine x squared minus food divided right idiots So we can plug in the village. Oh, sank in the hillbilly absented for here. So TV before in tow under 92 square minus four White idiot. Plus she&#x27;s not that pretty problem of the time The minute I saw the panel entirely tender nine X squared minus food bank for it. Bless you. So give in and dig it You in Vegas and given bouncer This is thank you

Shamshad Waris · Answer

Yes, it just divided by excess. Where you coming up? Nine minutes. Shooting that. We want a big substitution. Here. It&#x27;s up to you. It&#x27;s easy. Kitty. Sanjay Gupta, North end Very badly. There will say so. Yes, it&#x27;s a gun. I do most very better about Santa now 11. The video video X and B it in this integral. Then they will Yet in the place up here so we can actually cause kicked up into beauty divided by you. Write about it De san Kitta Get ahold of square Who come down Nine miners He, Sanjay Gupta with the whole issue How&#x27;s it behind this express on? I don&#x27;t even know it it in you off into digital divided by nine Sinus wet feet Doc into that is getting off now I think the doctor also dented This is so one day nine interview some hope I do know one by science Party medical Well said yes. Where that into beauty I do know the indignation of the process where technological minus four So we can I one day too into buying us or keep them now I live in the here we have it. Yes. After tooting sank a magical X by 80. So archetypical a week. And I did it by Uganda. Nine minutes. It&#x27;s just what? So you physical or in bus off? He resigned. Who? Come down nine minus x squared. No blood in the video off that. Here. My nose one day night walking 14 verse. Did he buy? Depend up nine minutes. Excess foot. I do know. I do know Court inbox. Parting parties must exit tickle. Wait. So here they get I want 1982. Did you conduct nine minus? It&#x27;s a square now, consumer kidding from a little better. And in a minute, then you get another answer. Minus one by tender bidding, you end up nine minus. It&#x27;s just what This is the antagonism given cancer using that romantic subject users. Thank you.

J Hardin · Answer

Let&#x27;s evaluate the integral of X squared over the square of nine minus X squared. Let&#x27;s use a trick substitution here since in the radical tyrannical expression insides of the form a squared minus X squared we&#x27;LL use the tricks of X equals a scientific data. In our problem, we have a equals three. So we have X equals three scientists. This is our tricks up. So we have the eggs is three coz I&#x27;m data. Also, we have the square root of nine minus six where let&#x27;s go to simplify this radical first this is nine minus nine sine squared. We could pull off the nine outside the radical that becomes a three one minus sign square on the inside. So the three square root of co sign squared, which is three co sign So this inner world becomes so X squared becomes nine signs where and then DX becomes three coats and data data and in the denominator we just worked that out over here, and that&#x27;s three co sign so we could cancel these three co signs. So we have nine into girl science. Where here We could use one of the Pythagorean or one of the true identities the double angle formula to rewrite this as one minus coastline of two theater, all divided by two. And then now we could integrate. So we have ninety eight over too, minus nine, signed to theatre number four and then plus our constant C of integration. So at this point, we could use the double ankle formula for sign to rewrite. This is to sign Data Co signed data with ninety eight over too minus nine Over too. Signed data cosign data plus e. So leave evaluated the new girl. But in order to get our answer back in terms of the original variable X, we should draw the triangle. So let&#x27;s go to the next patient. Do this. So our tricks up was X equals three signed data. So this means Scient Ada is X over three. So let&#x27;s draw any right triangle that has this property. So here&#x27;s data and then sign opposite over hypotenuse. So then we can use for that reindeer tto find the remaining side B. So we know B squared. Plus X squared is nine. So that B is equal to the square room of nine minus x work. So now we have all three sides of the triangle, So coming back to our original. So where we left off in the previous page, we had nine over Sue Daito minus nine over too. Signed data, coastline data plus e, not the data itself. We can get from our tricks up. So taking this equation up here, solving for data. So we take Sinan person both sides so we can write that for data. So we have a nine over too Sign in verse, eggs over three, minus nine over, too. And now sign Times Co sign. So sign is exploratory and co sign is be over three. So be is nine minus X squared inside the square root all over three. And then we can go ahead and cancel this nine with the nine in the bottom. And we still have a two in the bottom. And there&#x27;s a final answer

Amy Jiang · Answer

okay. Looking are square root. We feel that we have the form of we have X squared minus nine. Sooner of the form X physical, too. Nine is three squared. So that&#x27;s three. I think it&#x27;s a video. And that was taking our interpretive. That&#x27;s equal to three. A bit of a second is Chickens Canyon&#x27;s Okay, so let&#x27;s rewrite what we have. So we have the interval X is three seconds. Do you know that knowing chickens square, you know, over the square root of X squared cigarettes. Nine chickens square data minus nine. We contract out of nine and then we also have these chickens. There are tensions for the to accumulate this as in a row of nine times three chickens where Dido sequins better tangents, Barbera d data. And now we pull out that nine or the square root of line, which is three. And then you have a square root of seeking squared minus one, which is tangents. Square data. That&#x27;s that&#x27;s equal to tangent. They don&#x27;t. Okay, I know we can cancel our tangents of beta council this three as well and we&#x27;re left with nine Integral of Deakins squared. Vera, action. Is this you get cute of data? Better. Let&#x27;s see. Um, let me see if that&#x27;s cracked. Ticket squared chicken tangents, and we will have a lot of sinkings. Huge data. Okay. You know, we can use our A reduction formula, sophisticates. What was going to nine times, um, begins to power of three minus Tuesday. That sequins of data. And then we have changes there, over three, minus one. So that&#x27;s two. And then we have plus 1/2 in a row of sequins Data. I do know that the intercourse chickens Well, that&#x27;s equal to natural. Ah, sequence of data plus tangents. There on impulse. See? And then we&#x27;re seeking data intentions better too. Okay. You know, we get the following solution. You could rewrite this in terms of by drawing out or triangles.

Amy Jiang · Answer

Okay, so let&#x27;s start rewriting nine as three to depart to, and then we see that&#x27;s weaken. Rewrites are ex eyes. Um, X is equal to three tangents of data. And how d X will be able to three seeking square beta. Okay, so let&#x27;s replace our ex terms. So we get the square or the integral of nine tagine squared of data, and in the X is three took sequins squared data debido over the square root of We have nine and nine here so we can factor are discovered of nine, which is three. And then we have the square root of tendon squared, plus one kind of people to stick its squared of theta so we can get rid rid of Earth Square on our routes. And I wouldn&#x27;t cancel one of our sequins on or three. And we left with, um, the integral for nine, integral of tangents. Squared of data. I think it&#x27;s better. You better. Okay, so we want to check the integral falling. So let&#x27;s see Richard, you to be tangent. We need taken squared, but we don&#x27;t have that. So if we take you to be Deacon&#x27;s, okay, we&#x27;re gonna need a seeking and attention. Okay, I know. What&#x27;s your addition? In terms of signing coastline. So we have challenges, which is fine. Were they on? That&#x27;s over coast and square and sequences one over close science It is this coastline cubed. Do you deal And no Um okay, so actually, let&#x27;s go back a bit If use our identity for his Qianjin squared can rewrite that a sequence. So we commemorate this as, um you can squared of their own minus one and in time chickens data. You think I know we can expand this to get nine Grow up speaking cube better when it&#x27;s taking a bit off be better and then taking our fallen in a row Me yet nine times, um nine over to kind. Speaking of X, speaking of video tangents they don&#x27;t and then we have minus natural log of sequins of data plus tangents data and then we have crusty And now let&#x27;s find our triangles So we have X over three is equal to tangents of it, that is, we have a strangle X and three citizens the square root of X squared Post three squared, which is nine. So what&#x27;s no that&#x27;s begins a better. That&#x27;s people to, um what one of the coastline. Threat. Very over two square root of X squared plus nine and pretend INTs Orbital. We have X over three.

Problem

Evaluate the integral. $$ \int \frac{\sqrt{1+t^{2…

Problem 11 Medium Difficulty

Evaluate the integral.
$$
\int \frac{d x}{x^{2} \sqrt{9 x^{2}-4}}
$$

Answer

$\frac{\sqrt{9 x^{2}-4}}{4 x}+C$

Related Courses

Calculus Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 7

Video Transcript

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\frac{\sqrt{9 x^{2}-4}}{4 x}+C$

Calculus Early Transcendentals

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals