Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Evaluate the integral. $ \displaystyle \int \f…

04:48

Question

Answered step-by-step

Problem 7 Medium Difficulty

Evaluate the integral.
$\int_{0}^{a} \frac{d x}{\left(a^{2}+x^{2}\right)^{3 / 2}}, \quad a>0$


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

JH
J Hardin
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 3

Trigonometric Substitution

Related Topics

Integration Techniques

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Grace He
Heather Zimmers

Oregon State University

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

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.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

03:35

Evaluate the definite inte…

02:17

Evaluate the definite inte…

04:10

Evaluate.
$$\int_{0}^{2…

04:10

Evaluate.
$$\int_{0}^{2…

03:43

Evaluate the given definit…

01:43

Evaluate the definite inte…

Watch More Solved Questions in Chapter 7

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44

Video Transcript

we have the definite integral from zero to a of one over a squared plus X squared to the three half power where a is a positive number. So by looking at our denominator, we see a square plus x square, so we should take the tricks up to be X equals a tan theta. Therefore, D X's A C can square data the data. So observe here we have a definite rule, so we have two options on how to proceed. The first option is we can find the new limits of integration in terms of the variable data. If we do that, well, we won't have to draw the triangle because we won't need to go back to the variable X. The second option is to not find the new limits of integration. Then you would have to evaluate the anti derivative draw the triangle back to the Variable X, and then you could plug in these X values into the anti derivative. So let's see if we can take the first round so we don't have to draw the triangle here, so we need to find a new limits. So the lower limit and also before I go on. We have to make an observation here, and it comes from the tricks up. If you want to use this method, you have to be cautious about the the domain for data. So here, when we do this type of tricks of the T in substitution, your data lies between negative proper too, and powerful, too. So the lower limit. So before we had X equals zero. So plugging in X equals zero into this equation. Up here we have zero equals eight and data since a is positive number, that was the assumption. This means that tan has to be zero and the only time that tangent zero from negative pi over 22 pi over two is when data equals zero. So this is our new lower limit in terms of data. Similarly, for the upper limit, this was X equals a so plugging this into our tricks of equation. We have a equals a tan data, which means tan data is one and the only time that happens in this interval. Negative pi over 22 pi over two is when data equals power before mhm. So let's plug all this in. We have the integral from zero to power before DX becomes a C can squared data, data data. And in the denominator, we have a squared plus X squared, which is a squared 10 square data. And this is also the three house. Yeah, So at this point, let's take a look at this denominator here. And let's rewrite this before we before we go on. So here we can we can pull out of a squared, then we have one plus hand square data. So we have a squared Seacon square data to the three halfs and that will become a cubed times seeking cube data. So, yeah, now we could simplify. Let's cancel. So we we can cross off this a with one of those. So we have a square left over and we could cancel out those to see cans and then re elected with one second on the bottom. So we have one over a square, integral zero to pi before of one over. C can't data. Now. I'm running out of room here, so let me go to the next page One over a square, integral zero power before and then one oversee can't. By definition, it's called Santa. Yeah. So now we have we can evaluate this anti derivative. We know that science data and we have our endpoints. So now we just use the union circle to evaluate these. Sign of pyro for is square root 2/2 and then sign of 00 So there's our final answer.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
142
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Integration Techniques

Top Calculus 2 / BC Educators
Grace He

Numerade Educator

Heather Zimmers

Oregon State University

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

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.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
Recommended Videos

03:35

Evaluate the definite integral. $$\int_{0}^{2} x^{2}\left(x^{3}+1\right)^{3 / 2…

02:17

Evaluate the definite integral. $$\int_{0}^{2} x\left(x^{2}-1\right)^{3} d x$$

04:10

Evaluate. $$\int_{0}^{2} \frac{x^{3}}{\left(x^{2}+2\right)^{2}} d x$$

04:10

Evaluate. $$\int_{0}^{2} \frac{x^{3}}{\left(x^{2}+2\right)^{2}} d x$$

03:43

Evaluate the given definite integral. $$ \int_{0}^{2} \frac{2 x-1}{(x+3)^{2}} d…

01:43

Evaluate the definite integral. $$ \int_{0}^{1} \frac{x^{3}}{x^{2}-2} d x $$
Additional Mathematics Questions

00:53

'30 60 90 right triangle
The diagonal of a TV is 26 inches long: As…

01:30

'14x-9
8x + 3
If segment JK is congruent to segment KL, find the …

02:17

'If angle LMP is 11 degrees more than the measurement of angle NMP and …

03:35

'A farmer can grow two types of crop, X and Y. The profit per unit of X…

02:57

"can I have a solution to this problem?
Tests predict he has 0.85

02:50

'If line segment LK is congruent to line segment MK, LK=7x-10, KN=x+3, …

02:15

'3. If F, G, and H are the midpoints of the sides of AJKL, FG = 37 , KL…

02:15

'4. What trigonometric ratio would you use to find the distance from th…

01:55

'Is the triangle similar to △PQR? State whether each triangle is simila…

00:01

'Please answer my test. Please give accurate answer. Thank you!
Pro…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started