Definite Integral with Trigonometric Substitution

Evaluate the definite integral $\displaystyle \int_{9}^{18/\sqrt{3}} \frac{dt}{t\sqrt{t^2 - 81}}$ using trigonometric substitution.

This problem involves evaluating a definite integral using trigonometric substitution, a technique commonly used to simplify integrals involving square roots, especially those of quadratic expressions. The choice of substitution is often motivated by transforming the integrand into a form that is easier to integrate. In cases such as this one, where the integrand involves the square root of a quadratic expression, the substitution is typically one that relates to trigonometric identities, such as using sine or tangent to simplify the square root.

The key idea is to utilize trigonometric identities, for example, substituting expressions involving squares with sine and cosine functions can convert difficult algebraic forms into simpler trigonometric integrals. Specifically, for expressions like $\sqrt{t^2 - a^2}$, substituting $t = a \sec(\theta)$ is a common strategy, turning the square root into a readily integrable form (as $\sqrt{a^2 \sec^{2}(\theta) - a^2} = a \tan(\theta)$). Understanding which substitution to use can be as important as performing the integration process itself, as the right manipulation significantly simplifies the solving procedure.

Upon making the appropriate trigonometric substitution, the problem shifts into a more familiar territory involving basic integration techniques of trigonometric functions. Additionally, because this is a definite integral, careful attention must be paid to changing the limits of integration to correspond to the new variable introduced by the substitution.

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