# Use trigonometric substitution to evaluate the integral. {eq}\int\frac{\sqrt{x^2 + 36}}{9x^2} dx {/eq}

## Question:

Use trigonometric substitution to evaluate the integral.

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

## 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:

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Given Data:

• The given indefinite integral is {eq}\displaystyle \int {\dfrac{{\sqrt {{x^2} + 36} }}{{9{x^2}}}} \;dx {/eq}.

Apply the trigonometric...

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