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Integration by substitution, often referred to as u-substitution, is a technique to simplify the process of finding antiderivatives and integrals. Essentially, it involves changing the variable of integration to make the integral more manageable.

The method requires identifying a portion of the integrand that can be substituted with a single variable, typically denoted as 'u'. This piece is often chosen because its derivative is also present elsewhere in the integrand. After substituting, the integral is taken with respect to 'u', and once found, the result is re-substituted in terms of the original variable.

In our exercise, the substitution was made by setting u = tan(x+1), and the integrand was rewritten in a simpler form allowing the use of more straightforward integration techniques. This increased the clarity of the solution and made the subsequent integration steps more straightforward.

The secant function, denoted as sec(x), is the reciprocal of the cosine function, and therefore it is defined as sec(x) = 1/cos(x). It is one of the six fundamental trigonometric functions and holds significance in calculus, especially in integral calculus.

In the context of our problem, the secant function appears squared, written as sec^2(x), which is also the derivative of the tangent function. An important characteristic of the secant function when squared is that it appears in the integral form associated with the tangent function, such as in the derivative relationship d/dx [tan(x)] = sec^2(x).

Understanding this relationship is crucial when applying the integration by substitution method, as it allows us to recognize the potential for substitution using trigonometric identities, aiding in simplifying complex integrals.

The power rule for integrals is an indispensable tool in calculus. It states that the indefinite integral of a power of x is given by:\[\begin{equation}\int x^n dx = \frac{1}{n+1} x^{n+1} + C, \quad \text{where } n eq -1\end{equation}\]Here, 'n' is any real number except -1, 'C' represents the constant of integration, and 'x' is the variable of integration.

In our example, after performing the substitution and rewriting the integrand, we identified a square root, which can be expressed as a power of 'u'. The power rule is then applied to integrate u^(1/2), leading to the result (2/3)u^(3/2) + C. The final step involves reverting the substitution to express our antiderivative in terms of the original variable, x.

This rule is frequently used in solving indefinite integrals because it simplifies the process greatly, especially when the integrand is a polynomial or can be rewritten as one through substitution.