Use trigonometric substitution to evaluate the integral.

{eq}\int\frac{\sqrt{x^2 + 36}}{9x^2} dx {/eq}

Question:

Trigonometric Substitution Method:

In the trigonometric substitution method, we replace the integrand variable with a suitable trigonometric expression. It is also referred to as the integration by substitution, the reverse chain rule, u-substitution. The following integration formula helps us to solve the given indefinite integral problem.

$$\begin{align*} \int {{x^m}} \;dx &= \dfrac{{{x^{m + 1}}}}{{m + 1}} + C\\[0.3cm] \int {m \times h\left( x \right)} \;dx &= m\int {h\left( x \right)} \;dx\\[0.3cm] \int {\sec \left( x \right)} \;dx &= \ln \left| {\sec \left( x \right) + \tan \left( x \right)} \right| + C \end{align*} $$

Here, {eq}m {/eq} is the constant/numerical value.

Answer and Explanation: