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Evaluate

$$ \displaystyle \int \frac{x^2}{(x^2 + a^2)^{\frac{3}{2}}}\ dx $$

(a) by trigonometric substitution.

(b) by the hyperbolic substitution $ x = a \sinh t $.

A. $\ln \left(x+\sqrt{x^{2}+a^{2}}\right)-\frac{x}{\sqrt{x^{2}+a^{2}}}+C_{1}$

B. $\ln \left(x+\sqrt{x^{2}+a^{2}}\right)-\frac{x}{\sqrt{x^{2}+a^{2}}}+C$

Integration Techniques

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Let's evaluate the integral of X squared, divided by X squared, plus a square to the three has power. So for part A, let's use it tricks up. Since we have X squared plus a squared in the denominator in the radical our tricks up should be X equals eight and data and then taking the differential in East Side D. X equals a sequence where the data also let's simplify this denominator. So we have a squared chance where data plus a squares to the three halfs pulling out a square. Factoring that out, we have a square C can square data to the three halfs and then simplifying these exponents. We have a cube seeking cube Saito So are integral becomes we have X square, a pair of top in the numerator, so that becomes a squared chance for data. And then DX was a sea cans where, and we just simplified the denominator a moment ago and that they seek a cube seek and cute separate this from my earlier work. Now we see that we have a squared and another a and the top that's a cube and then a Cuban the bottom. So those go away. And then we could drop to the DC cans and we could take off two in the bottom. So we have one left over and then we have tangent squared data divided by C can't. So let me go to the next page here. I'm running out of room. So we originally had after simplifying, we had tan Square data, Oversea kin and many ways to proceed here. One of them might be to go ahead and just replaced hand with sine squared over Costa and square, and then use the fact that one over See Candice Co sign. So we have science for data over co sign and we could use the protagonist identity to rewrite. The numerator is one minus coastline square. And then we had in a roll C can data mine, isco's and data and the integral of sea can is natural log absolute value C can plus tangent and in a rule of co sign, is a sign now to find to evaluate all of these expressions in in terms of X, we go back to our original tricks up and we formed a triangle So our tricks of X equals eight and data implies that tan data is X over, eh? So if here's our data than opposite over adjacent is X over, eh? So we could put an X on the opposite A for the adjacent. If we call each the hype on news a squared equals X square plus a squared. So each equals radical Explore places where So now we can go out and evaluate seeking Tanja and sign. So see, Kanna Vato is each other, eh? So radical X squared plus C squared Oliver, Eh? Plus tan data which is exhilarating minus sign data and science. Ada is X over. H I'm running out of room. Let me come over here. It's the bottom left. We could clean this up a little bit so we could that we can find a common that I'm later which is a it Just add those fractions X squared plus a square. So now at this point, we already you can stop your But we could simplify this answer. So I'll just simplify. This is much as we can So that's our numerator divided by a Now the next step we can use the lot of property of logarithms that says natural log of be overseen his l N B minus lnc. So let it be the next step, minus natural log of a absolute value. And then for the sign we had. And now, at this point, we could just go ahead and replace Ellen A and Plus E with another constant those air. Just Constance. And this expression inside the absolute value is actually positive, even if X is negative x squared plus a squared plus eggs bigger than zero. And the way to see this is we could push the X on the other side, square both sides, and this is true. So that means that the original is true because our steps are reversible. So we just have natural log X squared, plus a square plus six minus X over X squared, plus a squared inside the radical plus de where we're taking D to be C minus Elena ve absolute value. So this is a simplified version for part a. Using the tricks up, let's go to the next case for part B. So for part B were to use the tricks up the hyperbolic trees up hyperbolic sign of teeth times a day. So let's recall the definition. So science h of hyperbolic sign of tea is even t minus e to the negative t over, too and, well, go ahead and take a derivative of this. So the related with respect to tea. Well, either that's he plus needed the minus t over to, and that's actually the definition of hyperbolic Cho sanity. Moreover, we have definitions for hyperbolic tannins and hyperbolic C Can't we'LL see In a moment we'LL need these. As you might expect, Tangent Ages define is hyperbolic. Sign over Hypermiling Coast and then hyperbolic c can is the finders won over hyperbolic osa. It can be shown using these formulas for sign agent cosign age that we have an expression that's very similar. The protagonist identity Not quite because we have a minus sign, but this is the closest thing to the protagonist in ity For signing hyperbolic Sinan Hyperbolic co sign again, you would show it by using these formulas for coastline inside. So this formula for co sign this formula for signed plug it all in square and simplify, And in our case, we'LL need a different version of this. So let's go ahead and divide both sides. Bye, hyperbolic co sign squared We get one minus hyperbolic stance Where equals hyperbolic C can swear and we can rewrite this There's ten hyperbolic stand squared equals one minus hyperbolic sequence Where Okay, we'LL see in a moment why we need all these So coming up here take the derivative differential on the side We just saw that the derivative of hyperbolic sinus hyperbolic co sign So it's coming from this Now we could write the integral that was given So we have X squared in the numerator So that becomes a squared, hyperbolic science where times the X just a hyperbolic cho sanity Dante. And in the bottom, we have x squared plus a square to the three halfs thought becomes a squared, sine squared age hyperbolic science squares plus a square to the three halves. Now let's pull out of a cube from the numerator and the denominator we could pull out of a square to the three halfs And then we're left with hyperbolic science weird plus one to the three house and using this formula on the right This means that science hyperbolic sine squared plus one is hyperbolic co sign squared. So this becomes hyperbolic Cose I squared to the three halfs, which is simply hyperbolic co sign to the third power and also a square to the three halfs. That's just a cute So we could cancel. Thank you. Never left over with hyperbolic science. Weird up top hyperbolic co sign all over Hyperbolic Oh, thank you. So in a moment we'LL see why will need this identity that'LL be the next step here because now we can cross off one of the co sides hyperbolic oh, signs We're left with two on the bottom coastlines where? So going to the next page We have the integral of hyperbolic tangent squared and using the Pythagorean theorem on page four, This is one minus hyperbolic C can't swear and the interval of one his team integral of hyperbolic c can't square this hyperbolic Sanja And then we have our constancy. So to show that the general of hyperbolic sea cans where his stance where there you can just go to the definition of hyperbolic tin hyperbolic she can And those come from the fact that we have formulas for hyperbolic Sinan Cho sun so recall from the earlier age. So we had These are the definitions of hyperbolic san and co sign since seek hyperbolic C Can't this one over hyperbolic co sign and then hyperbolic engine is hyperbolic. Sign over hyperbolic co sign. We can express all both of these in terms of even A T and e to the minus t and simplify. And then you can evaluate integral of seeking hyperviolent sequins were So now we would like to find then answer in terms of X so t we confined because our original tricks up hyperbolic tricks of excuse me was X equals a hyperbolic sign or tea. So that means hyperbolic sign of tea is eggs over, eh? So taking the inverse of hyperbolic sign on both sides. That's Artie. So we have sign hyperbolic sign in verse of and silver A And then we could write tangent. Hyperbolic tangent is hyperbolic. Sign over hyperbolic co sign and then we can go ahead and use defense. That sign a jj is eggs over, eh? So first, let me rewrite the sign in verse part Now for sign ht coming over here. Sign aged. He this exo grey. And now for the denominator hyperbolic co sign. We used the fact that from the identity involving hyperbolic coastline and sign solving for co sign gives us this, which is one plus X ray square, and we could simplify that as by getting a common denominator we could go on and rewrite. That is a squared plus X squared all over, eh? Plus he and now we're basically done the last step messages. Cancel that a's here in the second infraction. So let's go ahead and write that in the next line. And to spare our final answer. Hyperbolic sign inverse X ray minus X over Radical, A squared plus X Square plus C, and that's your final answer.