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JH
Numerade Educator

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Problem 44 Medium Difficulty

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

$ \displaystyle \int \frac{dx}{(1 + \sqrt{x})^2} $

Answer

$2\left[\ln (1+\sqrt{x})+\frac{1}{1+\sqrt{x}}\right]+C$

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Video Transcript

let's use a substitution toe. Rewrite this into grand as a rational function. So, looking at the denominator, let's try to take you to just be one plus square rex than taking a differential on each time and then using the power rule for derivatives we have This which we can also write is one over two times and I could replace Rue X using this equation. Bru X is just you minus one and we still have the X. So that was good. And multiply all sides of this equation. The left side on the right side, By the denominator, we have to you minus one. Do you equals DX so we can replace DX with this term over here and similarly looking at that denominator, including the square, We feel it. That's just use cleared from our substitution. So putting these two facts together this integral So we have dx up top. That's just too you might this one then we have to you on the denominator. We saw that to just be you square. Now, in this case, the partial fraction to composition. We actually don't need to go ahead and solve for the constants, because if you just write? This is two fractions. Then we could cancel one of the EU's and we just have too over you, minus two over you square. And that is our partial fraction. Could decomposition not just use the possible twice to evaluate first in a girl to natural log look and then for the second, using the power rule that's just becomes too over you and then plus our constancy of integration? The last step is to just replace you with X by using our original substitution. So here to natural log, let me go ahead and drop the absolute value since one plus rue exes, always bigger than zero, too over you and then plus E. And that's your final answer.