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Evaluate the integral$$ \displaystyle \int \frac{dx}{x^4 \sqrt{x^2 - 2}} $$Graph the integrand and its indefinite integral on the same screen and check that your answer is reasonable.
$\frac{\sqrt{x^{2}-2}}{4 x}-\frac{\left(x^{2}-2\right)^{3 / 2}}{12 x^{3}}+C$
Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 3
Trigonometric Substitution
Integration Techniques
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Let's evaluate the integral of one over X of the fourth times the square root of X squared minus two. And then we'LL graft too in a grand which is over here, circle in red will graft the indefinite integral which is after we evaluate this in a roll. Well, graph both of these on the same screen and will use this to check that our answer is reasonable. So first, let's go ahead and find that answer. Yeah, so, looking in the denominator, we see X squared minus two. So we should go ahead and take his air tricks up. X equals square root too. Seek and data, then taking a differential. Be a radical, too seeking timestamp data The so before we start plugging everything into the integral let's just go ahead and simplify this so extra the fourth, which we see in the denominator here that will be so using this equation up here, raise it to the fourth power and we should get four time seeking to the fourth power and then X squared minus two was good and simplify this so squaring X. We get to seek and square data and then we have minus two was fact around that radical too. You're left with C can't squared minus one. Use the Pythagorean identity for C can't intention and then take the square root of the square and you get radical to tan data. Okay, Now we're ready to plug our these all into the integral. So this integral and the numerator we see a DX. So we just replacing with this radical too Seek and data tan data data. And on the bottom, we have exit the fourth. We just simplified that that was force. He came to the fourth and then we also had this radical. We just simplified that a moment ago and it was radical to tan Taito. So as usual, the next best appears to go ahead and cancel is much as you can. It was radical. Two's go away, the tensions go away And you could even drop one of the sea cans here. So you have three left in the bottom. Then we still have that form the bottom. So let's go ahead and write this as one over four. We have won over C can. Cute. So that's right. This is Coach. Thank you. And then for this integral. We should go ahead and rewrite. This is co sign squared Times co sign and then rewrite The first co sign squared is one minus Sign squared Data using protagonist identity again. But this time for signing coastline. So I'm running out of room here. Let's go to the next page. The next step here will be to go ahead and take a use up. So let me copy that last expression down we had one fourth and then we'LL take a use of or any so after you sub Then we could go ahead and evaluate these. We have one over four. You might see you cute over three using the power rule. Plus he and then we'LL go ahead and replace you with Signe. So we have one over four signed data minus sign. Cute data over three. Plus he And now we've evaluated the general. Now the last step is to draw the triangle, cover everything back into the variable X. So coming back to our tricks up, we can rewrite it as c can't. Aita equals X over two. Let's try the triangle. Corresponding to this. Here's a date on the bottom, right, so c can is X over radical too. So that means high partner who's over adjacent is except for room two, the remaining side each. We could find it because it's a red triangle h where plus route to square equals X squared and then saw for each. And now we can evaluate Sinan sank you So this previously and rule over here this will become. Now let's go ahead and use a different color here one fourth. So sign data that's eight over eggs and then we have one third this three on the bottom and then sang cute. So we're going to take this previous expression that we just found and then raise it to the third power. So you get export minus two to the tree house, X cubed plus e. And if you like, you could go ahead and multiplied out for oh, and go ahead and right, this is so just cleaning up our answer a little bit. Here a t end multiplying the three and the four gives us a twelve on the bottom plus e. So this is the definite integral, which we're using now on blue. This is the definite into a girl So before we go to the graph, let's make an observation. So we've shown that let me use red for the Instagram. So we have the integral Then we had one over X to the fourth. This was the original problem. I'm using the color red to indicate that that's our immigrants. The graph corresponding to the in brain will be in red And then our final answer was in blue. And that corresponds to the blue graph that we'LL see in one moment. So we had one fourth. Let me actually clean this up a little yet. X squared minus two in the radical over four X That should be a radical up They're running out of room and then minus X squared minus two, three halves over twelve cubed and sense that we could choose any value for sea here When we drafted, we could just go ahead and take C to be zero when we use our graph because only will I need one graph. Let's make it easy by taking C v zero. So I wanted her point this out, sir. So we know what to look for in the graph. If this is true, then I could differentiate both sides, and I get the left hand side by the fundamental serum. The derivative will cancel out with the interval, so I just get the into grand left over and in the right hand side we have the derivative of the blue function so graphically, the red graph should be the derivative or the slope of the blue graph. And that's what we'LL look out for and temporarily when he grabs, It's So red hoops, Not plus E. We wanted the derivative of this blue graph, so you just tip the derivative of each side. So let's check a photograph of this matches up. So here the red graph, that's the inner end, and that should be the rate of change of the blue graph. So, looking at the raft here, for example, starting over here at about one and a half, we see that the blue graph has nearly a vertical tangent here and we would expect, and it's a positive slope. We would expect the rate of change there. The derivative is very large because it's almost a vertical tangent line. And the reason this is the reason that the road graph is so high at this point because of the riveted, the slope is very high. But as the blue graph is, ex starts increasing. We see the graph really starts decreasing. The slope starts getting smaller really fast. And so this is why the graph the red graph is falling downwards so quickly. And then at around two and onward, you could see that it looks like it. It stops growing or it's growing very slowly. So this is why the red Raph is getting closer to zero, and it's basically stabilising. If you want, you could come to the negative inside the third and fourth quadrants and make the same argument. So here, once again we have a vertical tension, and it's positive this is the same reason that the red graph is so large. And then the graph is still increasing from negative infinity up until about negative one and a half. But we can see that it starts basically get becoming flatter as we go in the negative direction. And this is why the red graph is getting closer and closer to zero. So our graph suggests that our answer is correct
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