Integration of sqrtx2 36 over 9x2

Integrate $\frac{\sqrt{x^2 + 36}}{9x^2}$ with respect to $x$.

To solve this integration problem, we need to employ a strategy that simplifies the integrand into a more manageable form. A common technique for such integrals is trigonometric substitution. This method is often used when the integrand involves expressions of the form $a^2 + x^2$ or $x^2 - a^2$. In our case, the expression under the square root is of the form $x^2 + a^2$, which suggests the use of a trigonometric identity to simplify the integral. Typically, we use substitutions like $x = a \cdot \tan(\theta)$ or similar trigonometric identities that can transform the integral into a simpler form involving trigonometric functions.

Once the substitution is applied, the integral often transforms into a form that involves trigonometric functions, which might be more straightforward to integrate. Following this, the integral can be solved through standard trigonometric integration techniques. In some cases, after integration, we substitute back the original variable to express the solution in terms of $x$ instead of the trigonometric variable. This is a powerful technique that highlights the importance of recognizing forms and patterns within integrals to employ the most effective integration strategy.

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