Evaluate the integral. int x sqrt(5x-1) dx

asked 2020-10-20
Evaluate the integral. \(\displaystyle\int{x}\sqrt{{{5}{x}-{1}}}{\left.{d}{x}\right.}\)

Answers (1)


The key here is to do a change of variables. Specifically, any time you see a square root, you either want to convert it to something squared under the square root or just set \(u=\) whatever is under it. In this case, we'll do the latter.
Let \(u=5x-1\).Then, because we know \(\displaystyle{d}{u}=\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}{\left.{d}{x}\right.}\), we see that \(du=5dx\) which is equivalent to \(\displaystyle{k}{\left.{d}{x}\right.}=\frac{{1}}{{d}}{u}\). Substituting these into the integral, we get.
This is part of what we want, but it still has that xx there and we want to totally convert to uu's. Well, remember that we set \(u=5x-1\). Let's solve that for x in terms of u:
Substituting this in, we get

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