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Question 11Integrate the function! trigonometric substitution121A4x2 1212{ hl R0 {121Ax2 _ 121hx...

Question

Question 11Integrate the function! trigonometric substitution121A4x2 1212{ hl R0 {121Ax2 _ 121hx

Question 11 Integrate the function ! trigonometric substitution 121 A4x2 121 2 { hl R0 { 121 Ax2 _ 121 hx



Answers

Using Trigonometric Substitution In Exercises $11-14$ , find the indefinite integral using the substitution $x=2 \tan \theta$

$$\int \frac{x}{2} \sqrt{4+x^{2}} d x$$

All right. So for this question, we start off the integral of two X squared DX, divided by four plus X squared the whole thing square, and we're going to use a substitution. That X is equal to two times attention to data so that D X is equal to three times as he can squared of theta d paid up. Now, substituting old that's in, you're gonna end up with an integral that looks something like this. Now, you should be able to simplify this integral down fairly easily by, um, expanding out the bottom notation So the bottom should be is off the form. April's beautiful squared, which is eight squared plus to a V plus B squared. Okay, so this is the This is what the denominator becomes. Onda, um you know, this is you have the numerator, which is that's year. That's eat times too, which is 60 eating times. Attention squared of data seek and squared of data dif ada using a trigger metric rules on events. These you should be able to solve this integral so that no one wants you back. Substitute X is equal to the tension that Feydeau you should be able to get 1/2 multiplied by the arc Tangent off X, divided by two minus two acts divided by four plus X squared. Plus, of course, you're

To integrate this. We will first rewrite us using completing the square. So from here we have the integral of one over the square of 36 -25 minus 10 X minus X squared. And then you have D. X. And this is just the integral of one over the square of 36 minus. Yes X plus five squared. And then the X. And then from here we will apply to econometric substitution. We want to let X-plus five equal to six signed feta. And differential of X will be equal to six casa in Sierra de Segura. So from here we have The integral of one over. We have 36 The Square of Express five will be 36 science squared theta. And then you have square here and then we replace the X by six. Go sign Sierra de Terra. Now this is just equal to the integral of six. Cosign Theta over. We have 36 squared times one minus sine squared theta and then squared. And then data. Now since one might assign squared fate is just go sine squared Theta. Then from here we have one over 216 times the integral of cosine Theta over Casa in Theatre. Race to the 4th power and then the data now we can reduce this to one over 216 times integral. Love one over cosine cubed data which suggests the same as seeking cube Sarah and then defer to So from here we have one over 216 times one half of 2nd X. or second data times tangent. Theta plus I'll. And absolute value of seeking thera plus tangent Theta. And then Vusi and this is equal to One over 432 of seeking theta times tangent. Theta plus L. And absolute value of seeking theta loss tangent Theta. And then plus C. Now since data is not the original variable then we go back to our substitution. That is if express five is equal to six sine theta. So then from here we have science data Equal to expose 5/6. And if we draw a right triangle with data here we know science theories opposite over the hypothesis. So this X was filed. This is six and the adjacent side by pythagorean theorem is equal to the square it of six squared minus Express five square that's just 11 -10 x minus x squared. So chic and terra which is equal to the hypothesis over the adjacent side. Yes, just equal to six over the squares of 11 minus 10 x minus x squared. And tangent. Theta will be equal to opposite over the adjacent side. That's equal to x plus five over the square of 11 minus 10 x minus x squared. So from here we have one over 432 times second Sarah which is six over the square of 11 minus 10 X minus x squared. This times tangent. Theta which is X. was five over the script of 11 minus 10 X minus X squared plus L. And absolute value of Second Thera which is six over scared of 11 minus 10 X minus x squared Plus Tangent Theatre which is experts five Over skirt of 11 minus 10 X minus X squared. And then plus C. Simplifying this. We have one over 432. This times six x plus 30 over the square it of 11 -10 X -X Square. That will be 11 -10 x X squared plus. We have L. N absolute value of six was X was five. That's express 11 over the square of 11 -10 x minus X squared and then plus C. So this is our indefinite integral.

So for this question, we start off with the integral of X cubed, divided by four times the square root off four plus X squared. And we have to use the substitution that acts is equal to two times the tension of data. So we have The tangent of fada is equal to X divided by two. So, uh, rewriting this we end up with eight times the tension cube off beta divided by four times the square root of four plus four times the tension squared off fatal so we can bring our Constance out to end up with two times the integral off the tension cubed of fada, divided by the square root off four plus four times the tension squared off data again. This could be further simplified to be two times the tension cube off data. Oh, that's just rewrite that tangent Cube off beta divided by We can factor out before that becomes a tubes. With this canceled out off the square root of one plus tangent squared off Fada. Now we know that sine squared plus co sign squared is equal to one dividing everything by Cho Science where we end up with tension squared plus one is equal to one over coast I lose is equal to the Seacat square. So therefore we have the tangent Cube Fada, divided by the sea can't squared of data. Now remember, the tangent is equal to sign divided by coast on. And now we're dividing this by, um, one over co science divided by one over coast or multiplied by co sign This has squared on it. This is a cubed and this is a cute So this cancels out of this so we end up but sign Cube Fada divided by co sign off Dana the data which is equal to the tangent of beta multiplied by the sine squared data. And as you can see that this just once you substitute this back into terms of X, this just ends up being four plus x squared to the power of three over two, divided by 12 minus the square root off four plus X squared. Plus, of course, are

In this problem, we have been asked to use a given formula from the table of integral to find a given indefinite integral. Now the formula that we have been asked to use the formula number 11 and that is one divided by U. Times E. Plus bu whole square D. You is equal to one divided by eight times one divided by E plus bu plus one, divided by E. Times the lawn of the modelers of you divided by E plus B. U. And with this we have C. Added C. Is the integration construct. Now the integral that we need to determine is one divided by X times two plus three X. Hold square V. X. So we can see from comparing these two expressions that the value of E will be too and the value of B will be three. Also instead of you we will have X. So we have one divided by A. So that's one divided by two times one by A plus B. U. So that we become two plus three X. Plus one by A. So we have one by two times one of you which will become extra wherever A plus B. U. So we'll have two plus three X. And with this we have the can't stand integration constant added as required, integral will be won by two times one by two plus three experts. Half times loan of the models have X divided by two plus three X. Plus the integration. Constancy.


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