Problem

Evaluate the integral using both substitutions $u…

08:40

Problem 30 Hard Difficulty

Evaluate the integrals.
$$\int_{0}^{2} x^{2} \sqrt{x^{2}+9} d x$$

Answer

$=\frac{\sqrt{2197}}{2}-\frac{9}{4} \sqrt{13}-\frac{81}{8} \ln \left|\frac{\sqrt{13}}{3}-\frac{1}{3}\right|+c$

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Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendental Functions

Chapter 6

Integration Techniques

Section 3

Trigonometric Techniques of Integration

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Video Transcript

so integrating from 0 to 2 of the function X square times, the square root of X Square plus nine. With respect to X, we can recognize that we have a sum of squares, so it's strong trying, right triangle. Insert enough data and let's build a right triangle for the relationship that would involve a sum of squares. We know by the vagrant there that if we have some of squares, the sum of squares would be in the would be the representation of the high pod news. The square root of the sum of squares, therefore, making each leg X and free. We can place the excellent three on either side. I'm recognizing, though, that each term of that perfect square we were take the square root of those to give us each leg. So now that we have built for right triangle, if we want a definition for X, we want to use the side X and the constant side in relationship to the data to build a dragon a metric equation. So in this case, we can see that tangent later is opposite over adjacent or, in other words, tangent. Data equals X provided by three multiplying both sides By three. We would obtain that X is three times tangent beta. Next we want to find BX, so DX would be the constant was stale. Three. The derivative a tangent, beta seeking squared fatal de beta. And technically, this is enough to build our integral in terms of beta bless. Go a little further and build the expression for the square root of X square plus nine. Still from our triangle. Simply want an expression for the square root of X square plus nine waiting to use that side along with the constant side and looking at the rich relationship toe data, we could use either co sign a data or secret data to involve the Jason side. In this case, I will choose the seeking pita, so recognizing that seeking data will be the high pot new side divided by the adjacent side, multiplying both sides by three who would obtain that the square root of X Square plus nine is three times seeking data. So now with the substitution XYZ, we can rewrite our integral in terms of data. As we do this, we want to also go ahead and change the limits of integration from X to data. So in the end, we don't have to go back to X. However, we very well could keep outward women's of integration as X. But in this case, let's recognize that when x zero use an hour Tanja Payday equation, we would have the tangent Data is equal to zero for three or zero. Therefore there would be tangent, inverse of zero and the tangent. The inverse of zero is zero. Likewise, when excess too. So likewise when excess to we would obtain that tanginess data is two divided by three. Or in other words, data equals the tangent in verse of 2/3. And instead of determining an approximation, pretension in person 2/3 will go ahead and use this as the exact value. Sorry answer. Growing X now becomes an integral data. Where when X was zero we got that are fatal. Was zero when X was too, he said that I were made. It was tension and verse of 2/3 x squared. It's now going to be free times. Tangent Beta wanted t squared, since sex was equal to three changes data and then next the square root of X square plus nine. We said Waas three times seeking data and then finally D X was three times seeking square data. De Pena next will rewrite the Inter role three square times three times three will give us an 81 hopeful that constant out in front tangent Square data and then times seeking to Peter with respect to theater. Next, we'll notice that Tangent Square data is equivalent to Seacon Square data minus one by the migrant identity. Slightly writing further, we now have seeking square beta minus one back. Want to two times he could cute Later. Deep data rewriting again Just distributing through. We now have secret to the death. Later minus You can cute later the data. So next weekend split, they enter grow into two inch girls, taking the integral seeking to the fifth beta separately from the control of seeking to Fada. But what I will do at this point just for space sake is less like a equal the integral of seek it to the fifth date of data. Let's let be equal the integral seeking cute data de data and work each of those integral separately and that will come back and substitute into our main problem. So computing these separately. We have a few options. We can either use a table integral or can use integration by parts. In this case, I'm going to use a table intervals and actually write the actual integral from the table that we would use in just a moment. But understand that if we were doing integration by parts, we would choose you to equal to power units less than the secret that's being shown, for instance, were seeking to the fifth later D data. We will let you equal seeking cute data and for seeking Cube think other integral There we will it you equal just seeking data. So again, too a power to less than what showing in the Inter Grand And then I was a result. But Devi would be what's remaining. So in the first integral for integral a the D V were people into Seacon Square Veda D data Come on TV and the be integral. But be let's remain in, which would be XY can't square Peter deep data as well. But in this case, we're gonna go ahead and use a table of intervals to compute each of these so I'm going to write the table up top. So, using a power reducing answer no formula for seeking to the end of data in our first integral of seeking to fifth data were use the power reducing formula with any cooling five. And the second case, we're gonna let in b three. So for interpret A, which is are integral seeking to the fifth date of beef data. If Ennis five we obtain wind about a five minus one just four kind, seek it to the power of five minus two, which would be three. They know tangent data plus and minus two be in the numerator. Well, says are Innis fi five minus two is three. The denominator will be in minus one since are satisfied five minus oneness for times the integral of seek it to the power of in minus two against us are in its five. In this first scenario, five minus two US. Three Been paid a deed data It will finish that peaks in just a moment. As you can see, we need the value for seeking cute later to finish off are integral a. So let's go ahead and compute are integral be using this same formula, but now within equaling three. So I have one divided by n minus one. Since our Innis three, we have three minus one, which is to time seek it to the n minus to the minus two. In this case, is one so secret to the first power fatum or just seeking data Times Tangent Data plus and minus two over in minus one. Since our Innis 33 minus two was one three minus one is to times that inter pro of seek it raised to the power of in minus two again, since our Innis three in this case three minus two is one. So we're just integrating seeking data with respect of data. Next will want to use another table of nickels for the integral seeking data. So using from are integral table that the integral seeking data Deke data is the natural ob ah, That's the value of secret data plus tangent data, plus the constant they now have for our integral be 1/2 time secret data changing data plus 1/2 times the natural log of absolute value of seeking data plus pages, data and it, plus our constant of integration until we substitute this expression back into our original problem. So now that we know what seeking cute later is equivalent to, we could finish off our answer for Integral A. We now have 1/4 time seeking cute Fada Tangent Data plus three force times are answer for seeking Cube data. So essentially, we're gonna take this entire answer for Big and insert it next to the three force. So the way I will do that for now is just a times B. And now we will substitute all of this information back into our original problem. So going back to our problem, we have 81. We now have integrated seek it to the fifth of data and we've also integrated seeking cube data. So now we're gonna have 81 times the expression that we obtained for a minus the expression that we obtain for B evaluated between zero in tangent in verse of two birds. So because of the length of our problem, we're going to have to carry out this the writing of the next step onto two lines. So let's begin writing now. So we'll start with 81 times the bracket And for the sake of these, let's go ahead and write in exactly what we have here for our be for a sorry so 1/4 seeking que pay the Times Tangent data plus three force times to be. And now, for the sake of having a little less writing for now, let's go ahead. And once we've inserted the A value, let's go ahead and then just subtract B without putting in the expression for P just yet. You are right that in a moment the soldiers were you two are writing just my bit because we'll notice that will be able to reduce some things before we write all of the larger expression out. So notice that within the bracket now we have the 1/4 time seeking cute data times, tangent data. But now we can quickly do a simplification on three force be minus B. So thinking of the minus B is being a minus one B, which is equivalent to thinking about a minus for four B. So we really have three force being minus 44 speed or negative 1/4 times our answer to our integral for B what this does for us now is it allows us to just right the expression for B once instead of writing it twice and then simplifying the larger expression. So now we have 81 times of quantity. Well, the 1/4 seek it. Huge beta tangent, beta stays. Then we have minus 1/4 times are entire expression for beaks, and we would change the color for B. And let's recall our expression for B was the 1/2 secret they had a tangent data, plus 1/2 times natural lot of absolute value of seeking beta plus tangent data so again, or be consisted of 1/2 time seeking data times Tangent data plus 1/2 times the natural log of the absolute value of sequence data plus tangent data. And instead of having R plus C, recall that you're bitter problem. We had our limits of integration because we had a definite into rule. So next less evaluate dysfunction at the upper limit of integration. So we have 81 times the quantity 14 time seeking cubed of tangent, inverse of 2/3 times tangent of tangent in verse of 2/3 minus. We'll go ahead and distribute the 1/4 brew as we re right so minus 1/8 time, seek it of tangent in verse. 2/3 times Tangent of Tangent Inverse of 2/3. Well, simple filing is momentarily continuing to distribute the negative 1/4 to the positive 1/2 we next half minus when a faith times the natural log of the absolute value of seek it a change in inverse of 2/3 class attain Geant battle the changing in verse 2/3 and the next. We want to subtract off what we obtained by plugging in or substituting in zero for all of the Vegas. But let's make a quick observation that when we've substituted zero that tangent of 00 so the first term will be zero tens of zero. Here would be zero as well tangent of zero p zero for this piece, but tension seeking of zero would be one, and we know the natural Aga one plus zero or the natural aka one would be zero. It's as a result, when we substitute in the lower limit, we would obtain a zero. Next, let's see how to compute the Sikh int of change in inverse or 2/3 to make our computations a little easier. Says recall that we had data was equal to tangent Anversa 2/3 therefore, tangent data was 2/3. So again, tangent data was 2/3. That means that we can now draw a right triangle. Now, as you build the right triangle based off the information that we have about change it, they'd have, um, 2/3. So if Tangent Data's 2/3 you know, the opposite side of that is to in the adjacent side, this three by the Bagram Thuram. We know that two squared plus three squared. I would give us four plus nine or 13 making the high pot new side square root of 13. As a result, we can now see that the seeking of data using the same triangle would be hype oddness over Jason or Square Root of 13/3. And since data was changing inverse of 2/3 squaring a 13/3 will represent the Sikh int of the Tangent in verse of 2/3. So finishing off our problem we now have 81 times the quantity 1/4 time Seek a cube Attention inverse of 2/3 were just computed the seeking of tangent in verse or 2/3 to be the square root of 13 divided by three. And now we want to cute that expression the tangent of tangent inverse of 2/3. Because of the inverse function property, with 2/3 falling within the right interval, that would give us just a 2/3. Next, we have minus one a. Again, the seeking of Tangin inverse of 2/3 were just computer to be a square root of 13/3. Again, the tangent of tangent inverse of 2/3 is again 2/3. Next, we have minus 1/8 times the natural log of the absolute value of seeking of tangent Anversa 2/3 which again was square root of 13/3, plus the tangent of the tension in first of 2/3 must go back and close off that absolute value sign. But the tangible change in inverse of 2/3 was 2/3 again closing off the absolute value sign those that offer bracket next. We need to simplify this further, and so as we continue to simplify, it comes down to the point of wanting to know. Do we want an exact answer or an approximate answer. So, of course, if we want an approximation, we can tight this expression into a scientific calculator to obtain the answer. For now, let's go ahead and simplify just a bit, Recognizing that we have won four times, Square root of 13 Cube would be 13 square root of 13. Three. Cubed is equivalent to 27. We still have the two times in three comfortably, right in a moment. Let's go ahead and simplify part of the second term. Two goes into itself one time to those into 84 times. So for the next term we have the square root of 13 in the numerator divided by four times three times three, which is equivalent to 36. Then we have minus 1/8 times the natural log of the absolute value of square root of 13 divided by three plus two, divided by three, which, because they have the same denominator, we could write PETA's one fraction as an option. Let's go ahead and simplify that first term a little further again. Recognizing some reductions, two goes into itself one time to goes into four two times, so now will define straight across. They have 13 times the square root of 13 in the numerator and 27 times six in the denominator, which would be 162 minus squared in 13/36 minus 1/8. Natural Log adapts that Value Square Reaper teamed by by three plus 2/3. We could go further and distribute the 81 we should see some more cancellations with the first terms denominators. But at this point, I'm going to leave the answer this way, recognizing that we can distribute the 81 reduce some more, or if we want the decimal approximation, we were obtained approximately 8.99 rounded to two decimal places.

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