Problem 31 Hard Difficulty

(a) Use trigonometric substitution to show that
$$ \displaystyle \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln (x + \sqrt{x^2 + a^2}) + C $$
(b) Use the hyperbolic substitution $ x = a \sinh t $ to show that
$$ \displaystyle \int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1} \left (\frac{x}{a} \right) + C $$
These formulas are connected by Formula 3.11.3.

Answer

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Video Transcript

Let's evaluate the integral of one over the square root of X squared, plus a square for part made, or to use the tricks up. Looking at the denominator, we see X squared plus a square. So we should take X to be eight times tangent of data. That's our trying. So and taking a differential in the side. We have DJ X equals eight time sequence. Where D data now in the denominator, we see X squared plus C squared inside the radical. So using our trips up, we could write. This's a square town square plus a square. We can faster out a square. And then we have Tan Square plus one, which is eight times the square root of sea cans where, which is a time C can't data. So our inner girl in our party becomes integral. We have DX that becomes a C can square data, and we just simplified the denominator. We did that over here and ended up being a times he can't fail so we could cross off those a's drop one of the sea cans and we just have integral seek and data D'Vera. Let's go to the next page. We still have integral seeking Tater d'Vera. And we know this integral to be the natural log in a rule. I'm excuse me, Natural log of seek and data plus tan data Now to evaluate seeking and tangent. We will need the triangle here. So let's go back to our original tricks up. X equals AIDS and data, which means Tien data equals X ray. So trying are right triangle here. Stato tangent is opposite over adjacent and we have that's equal to x ray. So we put an X there and a here Ages are about news Bye Protectorate and Iran. A squared equals X square plus a square So H equals to the radical of X squared plus days work. And now we can evaluate seeking attention That natural log c can't is high pollen news over and Jason so each other, eh? And the intention where we know that's Excell Ray from this and we could go ahead and combine those fractions. And then we can use the property of logarithms which states that Helen of x o ver y equals l N X minus Ellen. Why? So we use that it's natural log absolute value. Uh, I should have had a square here. Sorry about that. X squared plus a squared plus X minus, natural underway. Absolute value Bay plus e. So observe that this is just a constant. So we could go ahead and call this tea. And we could also observe that this expression inside absolute value is positive. And the reason for that is and swear places where is bigger than an equal to X. The reason for this is that if you square both sides, you have X squared plus a square bigger than equal to X squared. And that's true. So we can go ahead and drop the absolute value here in natural log explored, plus a square square root of that plus X plus and some constantly. And that's what we wanted to show him. Partner. Somebody check Mark. Here it is, indicate we're done with their and let's go to the next page Next page For part B. Barbie, we are to use X equals a times hyperbolic sign. So here's sign H, sometimes called cinch of tea. And this thing is defined. Bye. Signed age of tea is either the tea minus either the negative t over to If you differentiate it, We have either that he plus needed in the minus t over, too. And this is actually the definition of the hyperbolic co sign. So taking the differential on both sides of this hyperbolic truth sub the X equals a hyperbolic coastline of tea. So going back into the integral, let's also noticed that we'LL have to deal with in the inaugural, which was thie X Over Explore places where no DX is also whichever deeds here. Sorry about that. So that's the X and then on the bottom we have X squared. So that's a square sign, a squared of tea plus a square. Now we could pull out a here from the numerator and in the bottom. We could also pull out in a square from the radical and then we have sine squared sine age of square plus one. And now for this denominator, we'LL have to use an identity which is very similar to the pathology and identity. So using this for co sign and using this for sign, we could obtain the following identity hyperbolic co sign squared minus hyperbolic science Weird equals one. It's very similar to your usual protagonist identity for signing coast time. But we do have a minus on here, so it is a bit different. So using this, we can rewrite it. Sign a squared plus one. Evils cosign a square. So, in other words, the radical of signings Queer Plus One equals the radical of co sign. He's weird, which is just hyperbolic Cho sanity. So here we have a divided by a those cancel. Then we have hyperbolic co sign over hyperbolic Osa. Let's go to the next page. First, let's cross off those coastlines, hyperbolic coastlines. So we just have integral DT, which is just, I mean, in other words, with the integral of one dt. So we just have t plus e. And then remember, he is coming from our substitution. So we have hyperbolic sign equals X ray. So taking the inverse of both sides hyperbolic sign in verse, we can rewrite, he assign inverse sign age inverse of it's over. And this is the formula that we wanted for part B. So that completes part B of this problem, and there's a final answer