\(\int\left[\left\\{\left(x^{2}-1\right) d x\right\\} /\left\\{\left(x^{4}+3 x^{2}+1\right) \tan \left[\left(x^{2}+1\right) / x\right]\right\\}\right]=\ldots\) (a) \(\log \left|\tan ^{-1}\\{\mathrm{x}+(1 / \mathrm{x})\\}\right|\) (b) \(\log \left|\tan ^{-1}\\{\mathrm{x}-(1 / \mathrm{x})\\}\right|\) (c) \(\tan ^{-1}[\mathrm{x}+(1 / \mathrm{x})]\) (d) \(\tan ^{-1}[\mathrm{x}-(1 / \mathrm{x})]\)

Trigonometric substitution is a technique used in calculus to simplify integrals by replacing variables with trigonometric functions. This substitution is particularly useful when dealing with square roots involving quadratic expressions or when the integral contains expressions that resemble trigonometric identities.

For example, if an integral contains a term like \(\sqrt{x^2 + a^2}\), you might substitute \(x = a\tan(\theta)\) because \(\sec^2(\theta)\) turns the square root into a simple expression involving \(\sec(\theta)\). After substitution, the integral is usually easier to evaluate, often involving basic trigonometric integrals or other known integrals. Post-substitution, inverse trigonometric functions may be used to bring the results back into terms of the original variable.

Integrating rational functions, which are ratios of polynomials, often involves breaking them down into simpler pieces that can be integrated individually. One common approach is partial fraction decomposition, where the rational function is expressed as a sum of simpler fractions. This method is particularly effective when the degree of the numerator is less than the degree of the denominator.

Another strategy for integrating rational functions is to use substitution to transform the integral into a more manageable form. This approach was demonstrated in the step-by-step solution when the substitution \( u = x^2 + 1 \) simplified the structure of the integrand. By recognizing common derivatives within the function, one can often identify a suitable substitution that will aid in the integration process.

Inverse trigonometric functions, such as \( \arcsin(x) \) or \( \arctan(x) \) (also denoted as \( \sin^{-1}(x) \) and \( \tan^{-1}(x) \), are used to determine the angle that corresponds to a given trigonometric value. In calculus, these functions often show up after integrating expressions involving trigonometric substitutions.

For example, after using a trigonometric substitution, you might end up with an integral whose result involves \( \tan^{-1}(x) \). This inverse function allows you to express the solution in terms of the original variable. It's important for students to be familiar with the derivatives of inverse trigonometric functions, as these are often used in integration by substitution.

Integration by substitution, also known as u-substitution, is a method that involves changing the variable of integration to simplify the integral. The step-by-step solution provided is a perfect illustration of this technique. Here, identifying \( u = x^2 + 1 \) as a substitution helped transform the integral into a form that was easier to handle.

When applying this method, one must compute the derivative of the chosen substitution to replace the differential part of the integral (\(dx\)) with the new variable (\(du\)). If done correctly, the substitution often reduces the integral to a standard form or a simpler expression that can be integrated directly. Once the integral is evaluated, it's critical to substitute back in terms of the original variable to complete the solution.