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Problem 27 Easy Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\cos^{-1} (x^{-2})}{x^3}\ dx $

Answer

$-\frac{1}{2} x^{-2} \cos ^{-1}\left(x^{-2}\right)+\frac{1}{2} \sqrt{1-x^{-4}}+C$

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Video Transcript

Okay, This question wants us to integrate this function. So to do that, we want to transform this into the integral into one of our forms in the table. So to do this, let's make the substitution to make that transformation have. So we get you equal to X to the minus two. Because usually picking the argument of a coastline is a good start. So that makes d'you equal to negative, too. X to the minus three d x and we have X to the minus three d x. So we just gotta divide by negative too. So now we can transform are integral into negative 1/2 times the integral of co sign inverse of you, do you? And that's a much simply looking for him so we can look it up here, which gives us a formula for anti derivative. So let's just copy that down, keeping our negative 1/2 outside. So this is our anti derivative and now all we have to dio is substitute back in for you. So we get negative 1/2 times X to the negative to co sign in verse of X to the negative too minus square root of one minus X to the minus forth plus C and simplifying We get one over two X squared co sign in verse of one over X squared plus 1/2 square root one minus extra, the fourth plus C and that's your final answer.

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