[Math] Integral using table. What integration rule matches this integral

Question

I have this integral:

$$\frac{\sqrt{4 + (\ln x)^2}}{x}dx$$

Here are the relevant tables:

I think entry 23 is the closest one with the u on the bottom? But it's still not totally correct because of the $(\ln x)^2$ in the numerator. What can I usub?

EDIT

OHHH, I see, using the usub I get:

$$\int \sqrt{4 + u^2}du$$ where $u = lnx$

And I can use entry 21!

So:

$$= \frac{lnx}{2} * \sqrt{4 + ln(x)^2} + 2 * ln(lnx + \sqrt{4 + (lnx)^2} + C$$

Is that right?

Answer 1

Why dont you solve the integral instead. Notice that upon the subtitution $t = \ln x$, one sees that your integral is

$$ \int \sqrt{ 4 + t^2 } dt $$

Now, we may put $t= 2 \sec \alpha $ to obtain

$$ \int 2 \tan^2 \alpha 2 \sec \alpha d \alpha = 4 \int \tan^2 \alpha \sec \alpha d \alpha$$

can you finish ?