Question

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

Integrals
Evaluate the integral. $$\displaystyle\int{x}\sqrt{{{5}{x}-{1}}}{\left.{d}{x}\right.}$$

2020-10-21

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.
$$\displaystyle\int{x}\sqrt{{u}}\cdot\frac{{1}}{{5}}{d}{u}$$
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:
$$\displaystyle{5}{x}={u}+{1}\Rightarrow{x}=\frac{{{u}+{1}}}{{5}}$$
Substituting this in, we get
$$\displaystyle\int\frac{{{u}+{1}}}{{5}}\sqrt{{u}}\cdot\frac{{1}}{{5}}{d}{u}=\frac{{1}}{{25}}\int{\left({u}+{1}\right)}\sqrt{{u}}{d}{u}=\frac{{1}}{{25}}\int{\left({u}^{{\frac{{3}}{{2}}}}+{u}^{{\frac{{1}}{{2}}}}\right)}{d}{u}$$