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JH
Numerade Educator

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Problem 73 Medium Difficulty

Evaluate the integral.

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

Answer

$-\sqrt{1-x^{2}}+\frac{1}{2}(\arcsin x)^{2}+C$

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Video Transcript

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.