MHB What Is the Integral of sqrt(x-1)/x?

[MarkFL]

Here is the question:

What is the integral of sqrt(x-1)/x. Show Work.?


Answer should be 2sqrt(x-1) - 2arctan*sqrt(x-1) + C

Use substitution.

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

Hello livinginmyownreality,

We are given to evaluate:

$$I=\int\frac{\sqrt{x-1}}{x}\,dx$$

Using the substitution:

$$u=\sqrt{x-1}\,\therefore\,du=\frac{1}{2\sqrt{x-1}}\,dx\,therefore\,dx=2u\,du$$

$$u^2=x-1\implies x=u^2+1$$

And so we obtain:

$$I=2\int\frac{u^2}{u^2+1}\,du=2\int\frac{u^2+1-1}{u^2+1}\,du=2\int 1-\frac{1}{u^2+1}\,du$$

From this we obtain:

$$I=2\left(u-\tan^{-1}(u) \right)+C$$

Back-substituting for $u$, and distributing the $2$, we get:

$$I=2\sqrt{x-1}-2\tan^{-1}\left(\sqrt{x-1} \right)+C$$