sjeikdom0

2020-10-20

Evaluate the integral. $\int x\sqrt{5x-1}dx$

unett

The key in this situation is to adjust the variables. In particular, if you encounter a square root, you should either set it to something squared under the square root or convert it to something else $u=$ whichever is beneath it. We'll choose the latter in this instance.
Let $u=5x-1$.Then, because we know , we see that $du=5dx$ which is equivalent to . We obtain by substituting these into the integral.
$\int x\sqrt{u}\cdot \frac{1}{5}du$
This is a portion of what we want, but there is still a xx in there, and we like to completely switch to uu's. I mean, keep in mind that we set $u=5x-1$. Let's solve that for x in terms of u:
$5x=u+1⇒x=\frac{u+1}{5}$
By replacing this, we obtain
$\int \frac{u+1}{5}\sqrt{u}\cdot \frac{1}{5}du=\frac{1}{25}\int \left(u+1\right)\sqrt{u}du=\frac{1}{25}\int \left({u}^{\frac{3}{2}}+{u}^{\frac{1}{2}}\right)du$

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