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JH
Numerade Educator

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Problem 39 Easy Difficulty

Make a substitution to express the integrand as a rational function and then evaluate the integral.

$ \displaystyle \int \frac{dx}{x \sqrt{x - 1}} $

Answer

$$2 \tan ^{-1} \sqrt{x-1}+C$$

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Video Transcript

let's make a substitution such that after the substitution, this will be a rational function That means polynomial divided by polynomial. And then we'LL see if we can do maybe a partial fraction to composition. Maybe, Maybe not. Depends on the fraction. Here, let's try and take you to be X minus one in the radical. Yeah, or if you like, writers may be in one half, then, do you? We have one over to group X minus one D X. Now we can also write this, maybe push that to to the other side. And we have DX over radical X minus one so we can rewrite this so dx over the radical. That's just too, do you? But we have this extra ex term down here, but I can't write X here because, for example, I don't want to do that because I'm using a new variable You. So we have to lose the ex, come back to our substitution and solving for X so square both sides and that one over. And instead of writing X, we right, you squared plus one. So maybe this might be more recognisable if he plotted to. You may memorize this one. By now, if you haven't memorized it in the table, So let's maybe mention two ways to go about this. First, let's go ahead and just write that this is too ten and versatile. And the reason for this is because if you take the derivative attention inverse tangent, you may remember that from differential calculus. Or you could go ahead and actually just do it. Trips up here, you can let you be one tan data, and when you simplify, you should get ten in verse. Finally, let's get this answer back in terms of X so we just replace you by using the use of and that will be our final answer. So this equals to our ten radical X minus one, plus our constancy of integration, and that's our answer.