How to integrate the following ??

Question

#intdx/(1 - x^2)^(3/2)#

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Answer 1 · 2017-10-01T14:50:34

The answer is #=x/sqrt(1-x^2)+C#

We need #intsec^2x dx=tanx#

We perform this integral by substitution

Let #x=sintheta#, #=>#, #dx=costheta d theta#

#1-x^2=1-sin^2theta=cos^2theta#

#tan theta=sintheta/cos theta=x/sqrt(1-x^2)#

Therefore,

#int(dx)/(1-x^2)^(3/2)=int(costhetad theta)/(cos^2)^(3/2)#

#=int(d theta)/(cos^2theta)#

#=intsec^2theta d theta#

#=tantheta#

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