Integrate using Trigonometric Substitution

Integrate $\frac{\sqrt{4-x^2}}{x^2}$ using trigonometric substitution.

The technique of trigonometric substitution is a powerful tool in integration, particularly for integrands involving expressions like the square root of a sum or difference of squares. In these cases, trigonometric identities such as sin squared plus cos squared equals one can simplify integration by replacing complex algebraic expressions with trigonometric forms. For instance, substituting x equal to 2 times the sine of theta in the given integral can transform the integrand into a function of theta that is easier to integrate. This substitution exploits the Pythagorean identity to simplify the expression under the square root and remove it altogether after substitution. It's crucial to remember that, once the integration is completed, the result in terms of theta should be converted back to the original variable x using inverse trigonometric functions, where applicable. Additionally, trig substitution often requires using identities like the double angle formulas or identities involving secant, tangent, or cosine, depending on the substitution.

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