Evaluate the Definite Integral of x times the Square Root of x squared plus 9
Evaluate the definite integral .
To evaluate the definite integral of a function like over a specific interval, we need to consider which integration technique will simplify the process. Often, integrals involving square roots and polynomial expressions within them are suitable for trigonometric substitution. This technique leverages trigonometric identities to transform the integral into a more manageable form. Specifically, for this integral, the expression suggests a substitution that involves tangent or sine functions, given their properties that simplify square root expressions. By substituting , the integral becomes one involving trigonometric functions, which can be easier to solve.
Another approach is considering integration by parts if the structure of the integral suggests it, but here, trigonometric substitution is more direct. After performing the substitution, you will often end up with a trigonometric integral that requires simplification and application of fundamental trigonometric identities. After completing the integration in terms of the trigonometric variable, the final step involves back-substituting to revert to the original variable of integration. It's important to adjust the limits of integration according to the substitution made, or alternatively, to evaluate the indefinite integral first and then apply the original limits.
Understanding these high-level strategies for integration can significantly enhance your problem-solving skills. Integrals of this type test your insight into selecting appropriate techniques and executing transformations that are non-obvious yet powerful. This not only involves algebraic manipulation but also a comprehension of the geometric and trigonometric relationships that simplify complex integrals.
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