Question 1208255
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sqrt(1-x^2) is of the form sqrt(alpha^2-x^2) where alpha = 1.
We use the trig substitution x = sin(theta) which leads to dx = cos(theta)*dtheta


{{{sqrt(1 - x^2) = sqrt( 1 - sin^2(theta) )}}}
= {{{sqrt( cos^2(theta) )}}}
 = {{{cos(theta)}}}


In short,
{{{sqrt(1-x^2) = cos(theta)}}} based on the set up {{{x = sin(theta)}}}
Apply the reciprocal to both sides to determine that {{{1/(sqrt(1-x^2)) = 1/(cos(theta)) = sec(theta)}}}
A drawing of a right triangle might be helpful. 
{{{
drawing(400,400,-1,4,-1,5,
line(0,0,3,0),line(3,0,3,4),line(3,4,0,0),line(2.72,0,2.72,0.28),line(2.72,0.28,3,0.28),
locate(0.74,0.54,theta),locate(3.26,2.5+0.5,"opposite"),locate(3.26,2.5,"x"),locate(1.02-0.7,2.86-0.3,"hypotenuse=1"),locate(1.44-0.8,-0.32,adjacent=sqrt(1-x^2))
)
}}}
sine = opposite/hypotenuse
cosine = adjacent/hypotenuse
secant = hypotenuse/adjacent



Then,
{{{int(x/((sqrt(1-x^2))^3),dx) = int((sin(theta)*cos(theta)matrix(1,2,d,theta))/((cos(theta))^3),"")}}}


= {{{int((sin(theta)matrix(1,2,d,theta))/((cos(theta))^2),"")}}}


= {{{int((1/cos(theta))*(sin(theta)/cos(theta))matrix(1,2,d,theta),"")}}}


= {{{int(sec(theta)*tan(theta)matrix(1,2,d,theta),"")}}}


= {{{sec(theta)+C}}} Don't forget the plus C.


= {{{1/( sqrt(1-x^2) ) + C}}}


We therefore conclude
{{{matrix(1,3,int(x/((sqrt(1-x^2))^3),dx),"=",1/( sqrt(1-x^2) ) + C) }}}


You can use an online tool such as WolframAlpha to verify.
<a href="https://www.wolframalpha.com/input?i=int%28x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29%29">https://www.wolframalpha.com/input?i=int%28x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29%29</a>
The CAS mode in GeoGebra is another tool you could use.


Another way to verify is to apply the derivative to {{{1/( sqrt(1-x^2) ) + C}}}, and you should get {{{x/((sqrt(1-x^2))^3)}}}
I'll leave this for the student to do.



More practice is found here
<a href="https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx">https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx</a>
<a href="https://www.algebra.com/algebra/homework/Exponential-and-logarithmic-functions/Exponential-and-logarithmic-functions.faq.question.1208301.html">https://www.algebra.com/algebra/homework/Exponential-and-logarithmic-functions/Exponential-and-logarithmic-functions.faq.question.1208301.html</a>


Side note: There shouldn't be a space between the d and theta in the notation {{{matrix(1,2,d,theta)}}} but I couldn't get it to render without that space.
If the space wasn't there, then it would mistakenly render {{{dtheta}}}
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