How do you integrate $sqrt(1-x^2)$ ?

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Answer 1 · 2018-03-29T06:23:01

The answer is $=1/2arcsinx+1/2xsqrt(1-x^2)+C$

Let $x=sintheta$ , $=>$ , $dx=costhetad theta$

$costheta=sqrt(1-x^2)$

$sin2theta=2sinthetacostheta=2xsqrt(1-x^2)$

Therefore, the integral is

$I=intsqrt(1-x^2)dx=intcostheta*costheta d theta$

$=intcos^2thetad theta$

$cos2theta=2cos^2theta-1$

$cos^2theta=(1+cos2theta)/2$

Therefore,

$I=1/2int(1+cos2theta)d theta$

$=1/2(theta+1/2sin2theta)$

$=1/2arcsinx+1/2xsqrt(1-x^2)+C$

How do you integrate sqrt(1-x^2)sqrt(1-x^2)?