Integral of 1 Over Square Root of x Squared Plus 4 Using Trigonometric Substitution

Using trigonometric substitution, find the integral of $\frac{1}{\sqrt{x^2 + 4}}$ with respect to $x$.

Trigonometric substitution is a powerful method for evaluating integrals involving radicals, particularly those in the form of a squared term plus a constant, like we have in this problem. The basic idea involves substituting the variable in such a way that the new variable creates a trigonometric identity, simplifying the expression under the radical. For the integral given here, we focus on converting the expression under the square root into a form that leverages the Pythagorean identity. This often involves using a substitution such as x equals 2 tangent theta, which transforms the square root expression into a multiple of secant theta. The strategy ultimately simplifies the integration process by reducing it to a basic trigonometric integral, which can be more straightforward to evaluate. The method of trigonometric substitution not only aids in tackling integrals with radical expressions but also enhances understanding of trigonometric identities and their geometric interpretations. This approach is especially useful in situations where direct integration is impractical or leads to cumbersome algebra.

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