Chapter 5 - Section 5.5 - The Substitution Rule - 5.5 Exercises - Page 419: 69

$\int_{e}^{e^4}\frac{dx}{x\sqrt{ln~x}} = 2$

$\int_{e}^{e^4}\frac{dx}{x\sqrt{ln~x}}$ Let $u = ln~x$ $\frac{du}{dx} = \frac{1}{x}$ $dx = x~du$ When $x = e$, then $u = 1$ When $x = e^4$, then $u = 4$ $\int_{1}^{4} \frac{x~du}{x\sqrt{u}}$ $=\int_{1}^{4} \frac{du}{\sqrt{u}}$ $=2\sqrt{u}~\vert_{1}^{4}$ $=2\sqrt{4}-2\sqrt{1}$ $=4-2$ $= 2$