\(\int[(\tan \mathrm{x}) /\\{\sqrt{(\cos \mathrm{x})\\}]}=\) \(+c\) (a) \([(+2) / \sqrt{(\cos x)}]\) (b) \(-[1 / \sqrt{(\cos x)]}\) (c) \([(-2) /\\{3 \sqrt{(\cos x})\\}]\) (d) \([(-3) /\\{2 \sqrt{(\cos x})\\}]\)

Trigonometric integrals involve the integration of functions that contain trigonometric functions like sine, cosine, tangent, and others. These integrals often require specific techniques to simplify them into a form that's more manageable.

For instance, in our example \(\int[\frac{\tan x}{\sqrt{\cos x}}] dx\), the integral includes both tangent and cosine functions within a fraction. The complexity here is that there isn't a straightforward anti-derivative for this combination.

What's usually helpful is to rewrite trigonometric terms using identities, or, as in this case, to find a suitable substitution that simplifies the expression. The key is recognizing patterns, like a function and its derivative present within an integral, which can often indicate a viable substitution strategy. Here, noticing that \(\tan x = \frac{\sin x}{\cos x}\) allows us to see the integral's connection to the derivatives of trigonometric functions.

The substitution method is a powerful tool in integral calculus, particularly when dealing with complex trigonometric integrals. This technique involves replacing a part of the integral with a new variable, leading to a simpler expression that is easier to integrate.

In our exercise, the substitution \(u = \sqrt{\cos x}\) is chosen because it's related to the derivative of \(\cos x\), \(\sin x\). After finding the differential \(du\), we can replace both \(\sin x\) and \(\cos x\) with expressions involving \(u\), and importantly, \(dx\) with \(du\). This substitution transforms the integral into an expression in terms of \(u\) alone, \(\int -2 du\), trivializing the integration process.

The power of substitution lies in its ability to change the face of an integral from seemingly intractable to straightforward, but choosing the right substitution requires practice and insight into the functions involved.

In addition to the substitution method, integral calculus is full of various techniques to tackle complex integrals. Some common strategies apart from substitution include integration by parts, partial fraction decomposition, and trigonometric identities. Each technique serves a different type of integral, and sometimes they are used in combination to solve more complicated integrals.

For trigonometric integrals specifically, knowing trigonometric identities is hugely beneficial. These identities can sometimes simplify the functions within the integral to a form where a standard integration technique can be applied easily.

Understanding the conditions under which each technique thrives is essential for successful integration. For example, integration by parts is often used when an integral is a product of two functions, where one function can be easily differentiated and the other integrated. In complex scenarios, a deep understanding of these techniques can make a seemingly unsolvable integral manageable.