Verify the formula using any method.
![\[ \int \frac{\sqrt{a+bx}}{x} \, dx = 2 \sqrt{a+bx} + a \int \frac{dx}{x \sqrt{a + bx}} + C. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6a2213df98e1ecea6bdafbfbb40673a1_l3.png "Rendered by QuickLaTeX.com")
Proof. We multiply the numerator and denominator of the integrand by 
and then evaluate,
![\begin{align*} \int \frac{\sqrt{a+bx}}{x} \, dx &= \int \frac{a+bx}{x \sqrt{a+bx}} \, dx \\[9pt] &= \int \frac{bx}{x \sqrt{a+bx}} \, dx + \int \frac{a}{x\sqrt{a+bx}} \, dx \\[9pt] &= \int \frac{b}{\sqrt{a+bx}} \, dx + a \int \frac{dx}{x\sqrt{a+bx}}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6c1a414769eb851e77657b8853ded72e_l3.png "Rendered by QuickLaTeX.com")
For the integral on the left, we let 
, 
and we have
![\[ \int \frac{b}{\sqrt{a+bx}} \, dx = \int u^{-\frac{1}{2}} \, du = 2 \sqrt{u} + C = 2 \sqrt{a+bx} + C. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ef1b2b12509d72b822dc20d9b3e5c0dd_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[ \int \frac{\sqrt{a+bx}}{x} \, dx = 2 \sqrt{a+bx} + a \int \frac{dx}{x\sqrt{a+bx}} + C. \qquad \blacksquare \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b82e341d893fa9a2ef2427fee30e237c_l3.png "Rendered by QuickLaTeX.com")
Verify the formula using any method.
Proof. We multiply the numerator and denominator of the integrand by and then evaluate,
For the integral on the left, we let , and we have
Therefore,