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_0^{\frac{\sqrt{2}}{2}} \frac{x^2}{\sqrt{1 - x^2}}\ 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

University of Michigan - Ann Arbor

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.

06:13

02:53

Evaluate the integral.…

07:17

08:55

04:00

Evaluate the definite inte…

03:17

04:48

01:12

04:38

04:34

04:51

Find the Integral of \int …

Let's use a trick substitution for this integral because the denominator inside the radical is of the form a squared, minus X squared. I would go ahead and use X equal scientific era, then DX. It's because I'm dead so we can rewrite this integral. And we do have a definite on girl here, So we do have the opportunity or not necessary. It's not required, but it can simplify your work to rewrite these limits of integration in terms of fatal if possible. So here plug in X equals zero and then recall when we do it. Troops up of this form signed data you require. That data is in between negative, however too entirely too. So the only solution to this equation and dishonorable up here is state a equal zero. So our new lower limit will still be zero and then plug in the upper limit for X in. The only solution in this interval that makes this possible is PIRA for so that's our new upper limit. And then we see X Square up here in the numerator, so that becomes science squared after the substitution and then we have DX. So that's co sign D data. And then in the denominator, we have one minus X squared inside. The radical selections come to the side to evaluate this and then use the protector in Toronto for sign and co sign to write. This is co sign squared, and then you can evaluate. This is just cosign data. So that's your denominator. And then we could go ahead and cross off those one half. So here we're left with just pie before and then we have just signed squared only. So here I would use tohave angle formula for science. Queer. That's one minus co sign to date over, too. And then here I'm just pulling out the one half and then I have one minus course I into data left over. So that's one half. Now it's in a very that's data minus scientist eight over, too zero pyro for. And then let's just go ahead and plug those and points in so plugging Piper for first. So we have a pilot or for minus sign pi over too, over two, and then when we plug in zero both terms or zero, we have zero minus sign zero over to, so there's nothing to subtract and then sign a pie over to is just one. So we just have pav rate. So we write this pie over eight and then minus one over four, and this is after we did distribute the one half, and that's your final answer.

View More Answers From This Book

Find Another Textbook

01:39

Read each item carefully and write the correct answer on the blank provided …

00:51

Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more …

00:27

Jenny went to school for 20.5 days in a month. she bought lunch for php 45.8…

02:41

5.) Lope harvested 163.5 kilos of pomelos while Marcon harvested 362.75 kilo…

02:40

The Hair Company offers two brands of shampoo: brand H and brand S.100 p…

01:20

Evaluate 3 + 6 x (5 + 4) ÷ 3 with explanation

02:02

780 students stand in rows and columns. Each row has an equal number of stud…

02:09

1. Give the domain Of Y=√4 - x² using set builder notation.

01:03

38. The mass of a rectangular sheet of a wood varies jointly as the length a…