Evaluate Integral of Secant Cubed Using Trig Substitution

Evaluate $\displaystyle \int \sec^3 \theta \, d\theta$ using trig substitution where $x = \tan(\theta)$.

This problem involves evaluating an integral, specifically the integral of secant cubed in terms of theta, using trigonometric substitution. Trigonometric substitution is a technique particularly valuable in calculus when dealing with integrals involving square roots that are not easily simplified by basic algebraic manipulation. It utilizes relationships and identities from trigonometry to simplify the integrand, making the integral more manageable or even straightforward to solve.

When solving this problem, the target substitution is one that simplifies the term involving the secant function. Here, the substitution of $x = \tan(\theta)$ can be beneficial because it relates the trigonometric function secant to an expression in terms of tangent. This substitution will likely lead to simplifying the integrand into a more recognizable or integrable form.

The process of trigonometric substitution not only involves setting up the substitution but also requires changing the differential element (in this case, d theta) and carefully managing any resulting bounds if the integration was definite. This practice helps develop a deeper understanding of the intricate connections between trigonometry and calculus and is an essential tool for solving more complex integrals involving trigonometric functions.

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