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

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Answer 1 · 2017-06-18T01:23:05

Use parts, or substitution and then parts.

$int cos^-1 x dx$

Let $u = cos^-1x$ so $du = -1/(sqrt(1-x^2) dx$ and

let $dv = dx$ , so $v = x$

$uv-int vdu = xcos^-1x + int x/sqrt(1-x^2) dx$

Use substitution $u = 1-x^2$ to finish with

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

OR replace the inverse cosine first

Let $theta = cos^-1x$ so that $cos theta = x$ and -sin theta d theta = dx#

The integral becomes

$-inttheta sin theta d theta$ Which may be integrated by parts with $u = theta$ and $dv = sin theta d theta$

When finished we get

$- (sin theta - theta cos theta) + C$

And reversing the substitution gets $xcos^-1 x - sqrt(1-x^2) +C$ as above.

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