How do you find the integral of $cos^(-1)x dx$ ?

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Answer 1 · 2016-09-07T14:28:59

$=x cos^(-1) x - sqrt(1-x^2) + C$

$int cos^(-1) x dx$

we know $d/dx ( cos^(-1) x ) = -1/sqrt(1-x^2)$ so we can try set up an IBP and use that fact

$= int d/dx (x) cos^(-1) x dx$

$=x cos^(-1) x - int x d/dx ( cos^(-1) x ) dx$

$=x cos^(-1) x + int x*1/sqrt(1-x^2) dx$

we know that $d/dx (sqrt(1-x^2)) = 1/2 1/sqrt(1-x^2) (-2x) = -x/sqrt(1-x^2)$

$=x cos^(-1) x + int d/dx(-sqrt(1-x^2)) dx$

$=x cos^(-1) x - sqrt(1-x^2) + C$

How do you find the integral of cos^(-1)x dxcos^(-1)x dx?