Integration of Square Root Expressions

Integrate the square root of $2-x^2$ over $x^2$.

Integrating square root expressions often involves the technique of trigonometric substitution, which transforms the integrand into a trigonometric function that can be more easily integrated. In this problem, the expression under the square root sign suggests using a substitution that will simplify the expression into a standard trigonometric form. This is typically done by recognizing patterns that resemble the Pythagorean identities.

For the given integral of the square root of $2-x^2$ over $x^2$, one common approach is to use a substitution where x is replaced with a trigonometric function, such as sine or cosine, to exploit the identity $\sin^2\theta + \cos^2\theta = 1$. This substitution simplifies the square root expression, allowing for a trigonometric integration method to be applied. The process requires careful manipulation of the integral limits and the substitution itself to ensure the integral is expressed correctly in terms of the new variable.

Once the substitution is successfully applied, the integration becomes straightforward, following recognition of standard integral forms of trigonometric functions. After integrating, it is crucial to back-substitute to return to the original variable. This problem not only tests your skill in performing integration but also enhances your understanding of trigonometric identities and substitutions, an important concept in calculus.

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