Integral Evaluation Using Trigonometric Substitution2

Evaluate the integral $\displaystyle \int \frac{1}{x^2 \sqrt{x^2+4}} \, dx$ using trigonometric substitution.

Trigonometric substitution is a powerful integration technique used to simplify integrals that involve square roots of quadratic expressions. In this problem, the presence of the square root in the denominator suggests a substitution strategy that involves introducing trigonometric identities to facilitate integration. This method usually involves substituting a trigonometric function for a variable, transforming a difficult algebraic integral into a more manageable trigonometric integral.

Consider the expression under the square root, $x^2 + 4$, which matches the form $a^2 + x^2$. This is a hint that an appropriate trigonometric substitution would involve the tangent function since the identity $1 + \tan^2(\theta) = \sec^2(\theta)$ directly relates to such forms. By letting $x = 2\tan(\theta)$, the integral can be rewritten in terms of $\theta$, significantly simplifying the process. Additionally, recognizing the derivative chains and simplifying the resulting trigonometric forms are crucial steps in this technique.

Using trigonometric substitution not only streamlines the integration process but also reinforces the understanding of trigonometric identities and their applications in calculus. It’s essential to follow through with proper back substitution to return the final answer to the original variable, ensuring that the process is complete and precise. Mastery of these techniques is valuable, as they are applicable to a wide range of problems involving square roots and rational functions in integrals.

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