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Evaluate the integral.

$ \displaystyle \int \frac{x + \arcsin x}{\sqrt{1 - x^2}}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Missouri State University

Harvey Mudd College

University of Nottingham

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.

02:15

03:10

Calculate the integrals.

05:20

01:38

Evaluate the integral.…

01:05

00:46

Evaluate the indefinite in…

01:10

Calculate.$$\int \frac…

01:32

00:40

02:18

04:00

01:34

let's start this one by just putting it into two fractions here. Or even better, let's just right is to separate animals completely. So in the first line we have except top. And then for the second one, we have that arc sine and the other seemed denominator. Now let's do some substitution is here for the first one. Let's take you to just be that denominator. Actually, let's take it to be just the expression inside the radical, then hear negative do you over to will just give you extra e x And then for this one Over here, Let's take a w to be orc sign then DW is one over the square of one minus x square D x. So we're doing useful for both of these. So for the first Indian girl, we have this negative one half coming from the day you took and then that takes care of the ex DX. So we just have a d you up here and then we have a square of you in the bottom. So, Lucious, right? That is you to the negative one half and then we have plus and a girl w and then dw Now let's just use the Powerball twice here. So for the first integral, that's Debbie to the one half. And then you divide by a half and then for W have any squared over two. It's ad that constancy. And so here you could cross off the one half with the two. You have a negative here, so negative square of you and then replace you with X so one minus X square in the Radical plus. And then here you have W Square. So that signed inverse of X all square and then divide that expression over too, plus our constancy there at the end, and that's my final answer.

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