Answer
$$ - \frac{3}{{11}} \cdot \ln \left| {\frac{{11 + \sqrt {121 - {x^2}} }}{x}} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{3}{{x\sqrt {121 - {x^2}} }}} dx \cr
& or \cr
& = 3\int {\frac{1}{{x\sqrt {{{\left( {11} \right)}^2} - {x^2}} }}} dx \cr
& {\text{integrate by tables using the formulas on the apendix D for this book}} \cr
& {\text{using the formula 9}}:\,\,\,\,\,\,\,\int {\frac{1}{{x\sqrt {{a^2} - {x^2}} }}dx = - \frac{1}{a} \cdot \ln \left| {\frac{{a + \sqrt {{a^2} - {x^2}} }}{x}} \right|} + C \cr
& {\text{in the integral }}\int {\frac{1}{{x\sqrt {{{\left( {11} \right)}^2} - {x^2}} }}} dx{\text{ we can see that }}a = 11 \cr
& 3\int {\frac{1}{{x\sqrt {{{\left( {11} \right)}^2} - {x^2}} }}} dx = - \frac{3}{{11}} \cdot \ln \left| {\frac{{11 + \sqrt {{{\left( {11} \right)}^2} - {x^2}} }}{x}} \right| + C \cr
& {\text{simplifying}} \cr
& = - \frac{3}{{11}} \cdot \ln \left| {\frac{{11 + \sqrt {121 - {x^2}} }}{x}} \right| + C \cr} $$
