Integration of Rational Expression - Alternative Method

Hello everyone,

I integrated the following rational expression correctly, but one of my steps differ from that one of the book. Could anyone tell me if what I did is still correct?

Thank you!

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1. Integrate: $\displaystyle \int \frac{\sqrt{x - 4} + x}{x - 4} dx $

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$\displaystyle \int \frac{\sqrt{x - 4} + x}{x - 4} dx $

= $\displaystyle \int (x - 4)^{-1/2} dx + \int \frac{x}{x - 4}dx $

Let $\displaystyle u = x - 4 $ and $\displaystyle du = dx $. Therefore, $\displaystyle x = u + 4$.

= $\displaystyle \int u^{-1/2} du + \int \frac{u + 4}{u}du $

= $\displaystyle 2\sqrt{x - 4} + \int \frac{u}{u} du + \int \frac{4}{u} du$

= $\displaystyle 2\sqrt{x - 4} + x + \ln(x - 4)^4 + C$

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However, my practice workbook integrated $\displaystyle \int \frac{u}{u} du $ as $\displaystyle u $, and then substituted $\displaystyle x - 4 $ back into the equation.

So according to the workbook, its last step would be:

= $\displaystyle 2\sqrt{x - 4} + u + \ln(x - 4)^4 + C$

= $\displaystyle 2\sqrt{x - 4} + (x - 4) + \ln(x - 4)^4 + C$

= $\displaystyle 2\sqrt{x - 4} + x + \ln(x - 4)^4 + C$