Download the App!
Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.
Question
Answered step-by-step
Evaluate the integral.
$ \displaystyle \int \frac{1}{x^3 \sqrt{x^2 - 1}}\ dx $
Video Answer
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Like
Report
Official textbook answer
Video by J Hardin
Numerade Educator
This textbook answer is only visible when subscribed! Please subscribe to view the answer
Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 5
Strategy for Integration
Integration Techniques
Campbell University
Oregon State University
Harvey Mudd College
Idaho State University
Lectures
01:53
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
27:53
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
03:52
04:18
Evaluate the integral.…
03:29
01:04
Evaluate the indefinite in…
01:15
Evaluate the definite inte…
01:48
02:39
02:22
Evaluate the integrals…
0:00
04:51
Find the Integral of \int …
to evaluate this inner girl by trying to trick substitution X equals seeking data and the motivation for this This choice of a treat so is based on not necessarily the radical, but more so the term X squared minus one. So this is a number of the form X squared minus a square. And when you're in this case and you want to use shrinks of, then this is the tricks of that you should use. So here, differentiate. So that's already inside. And then now we can rewrite this integral So dj X up top. I can replace that with this. So I have seeking data tentative data up top and then on the bottom C can cube, and then we'LL have square root seek and square data minus one. Now, here we know that seek and square data minus one equals stance. Where there are this's one year Pythagorean identities. So when we take the square room, we just get tan data and so we can go ahead and cancel this tan data here with this tan date up there and then we could cross off this. He can't with one of these and we have two left on the bottom. So we have one over c can squared and we could rewrite that as the integral of co signed square. Really? No. And then used a half angle identity for co sign. And we could integrate both of these terms If this through data is bothering you feel free to do use up here. You can do u equals to data and we have one half data and then we have sign to theatre and we'll have a to from this over here and then we'LL get another two from the use of And then here we can rewrite this using the double angle formula for sign. And then we could cross off that too, with one of the other two's. So we have one half data. So let me just write that a state over to and then signed Data Co signed data over two plus c. Thanks. Now let's evaluate each of these three terms, and these will all come from our use of So, for example, if I take in verse, he can on both sides I find data so I can rewrite this answer as C can't members of X over to you urgent. And then now I need to evaluate, sign and co sign. So this is where I go to the triangle. So we did seek and data equals X over one. So that gives us X divided by one. If recalling this angle over here data no. And then by Pythagorean dirham, we get the remaining side. So now we could go ahead and find sign Ako side. So, science officer, over half pound news. So we have over X for sign and then for co sign Jason over. Heaven knows it's one over X and then we still have our two down there and the last appears Just simplify this answer a little bit. She can inverse effects over too square root, X squared minus one over to X squared. Plus he and that's our answer.
View More Answers From This Book
Find Another Textbook
02:11
Lucas weighs 200 pounds. He starts a weight-loss program that helps him lose…
The probability of picking two black marbles from a box at random without re…
00:49
Janke’s garden is in the shape of a right isosceles triangle with base 3.4 m…
00:28
A gas pump measures volume of gas to the nearest 0.01 gallon. Which measure…
01:40
Brian is solving the equation x^2 - 3/4x= 5. What value must be added to bot…
You have just graduated from college and purchased a car for $8000. Your cre…
Forty-five basketball players participate in a tournament. Write and solve a…
02:12
joseph and patrick purchase school supplies in the school bookstore. joseph …
01:30
A set of data points has a line of best fit of y = –0.2x + 1.9. What is the …
Which function increases at a faster rate on 0 to infinity, f(x) = x2 or g(x…