Evaluate Integral Using Trigonometric Substitution23456

Evaluate the integral $\displaystyle \int \frac{dx}{\sqrt{x^2 + 2x + 145}}$.

In this problem, you'll be evaluating an integral that involves a quadratic expression inside a square root in the denominator. These types of integrals often require the use of trigonometric substitution to simplify the expression and make the integration process more straightforward. Trigonometric substitution is a powerful technique used in calculus to transform integrals involving square roots of quadratic polynomials into more manageable forms.

The key concept here is to recognize when a trigonometric substitution is applicable. Typically, a substitution like this is useful when you have expressions that fit the patterns of a squared variable plus or minus a constant, or could be rewritten in this form. Once you've identified this, the goal is to choose a substitution that simplifies the radical expression. For instance, with an expression like $x^2 + a$, a common approach is to use the substitution $x = a \tan(\theta)$, which transforms the integral into one involving trigonometric identities that can be more easily integrated.

Once the substitution is made, the integral often translates into a form that is easily solvable using standard trigonometric integrals or further simplifications. Understanding how to transition back from the substitution form to the original variable is also crucial, as it involves using inverse trigonometric functions and often requires some algebraic manipulation to simplify the expression. This problem illustrates the practical application of using appropriately chosen substitutions to tackle seemingly complex integrals efficiently.

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