Integral of 1 over x squared plus 4 using Trigonometric Substitution

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

The integral of the form 1 over x squared plus a constant can often be evaluated using trigonometric substitution, a technique in calculus that simplifies integrals by substituting trigonometric functions for algebraic expressions. This method is especially useful when dealing with quadratic expressions under a square root or in the denominator, as in this problem. The key idea is to transform the algebraic expression into a trigonometric identity that is easier to integrate.

For instance, in this problem, by recognizing the expression under the integral as a form related to tangent or cotangent identities, you can use a substitution that leverages these identities to simplify the integral. When applying trigonometric substitution, it's crucial to carefully choose the substitution based on the structure of the quadratic expression. Here, recognizing that x squared plus 4 can be associated with a tangent substitution involves setting x equal to 2 tangent theta, capitalizing on the identity 1 plus tangent squared theta equals secant squared theta.

This substitution simplifies the integral into a form that can be easily integrated using basic trigonometric identities and reverse substitutions. After carrying out the substitution and integrating, the final step involves substituting back to the original variable, which may require additional algebraic manipulation to express your result in a clean, algebraic form. This technique is an insightful example of how trigonometric identities can be used to tackle integrals that at first glance do not seem to involve trigonometric functions at all.

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