91Ó°ÊÓ

When tackling definite integrals that do not fit standard formulas, integration by substitution proves to be a powerful strategy. Similar to the u-substitution method used in indefinite integrals, it involves changing variables to simplify the integral.

For example, consider an integral with a denominator of the form \(1 + 4x^2\). A substitution \(u = 2x\) simplifies this to \(1 + u^2\), which is more straightforward to integrate. It's essential to also adjust the differential; since \(du = 2dx\), we compensate by including a factor of \(\frac{1}{2}\) outside the integral. Don't forget to change the limits of integration to match the new variable, \(u\). This method streamlines the integration process, leading to a solvable integral in terms of \(u\).

Recognition of common integral forms is crucial for solving integration problems efficiently. The arctangent function, denoted as \(\text{arctan}(x)\) or \(\tan^{-1}(x)\), is frequently encountered in calculus.

It serves as the inverse of the tangent function, confined to values between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) where it's bijective. In integration, if you encounter a derivative of the form \(\frac{1}{1+x^2}\), you can directly link it to the derivative of \(\text{arctan}(x)\). In our current problem, the integral after substitution closely resembles this form, allowing for the arctangent function to provide the antiderivative necessary to evaluate the integral.

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, representing the heart of calculus. It consists of two parts, with the first indicating that the integral of a function \( f \) over an interval can be calculated using any of its infinitely many antiderivatives.

The second part entails the evaluation of a definite integral. Once an antiderivative of the integrand is known, the definite integral from \(a\) to \(b\) is found by taking the difference of the antiderivative evaluated at \(b\) and \(a\). This principle allows us to turn the integral \(\frac{1}{2}\text{arctan}(u)\) into an easily solvable expression by plugging in the new limits of integration.

In the antiderivative evaluation step, we need to use what we've learned to find the indefinite integral, which we will then evaluate at the bounds provided. After identifying the arctangent, the respective antiderivative of \(\frac{1}{1+u^2}\) in the context of our problem is simply \(\text{arctan}(u)\).

When we apply the Fundamental Theorem of Calculus, we plug in the upper limit of integration, subtract the derivative evaluated at the lower limit, \(\frac{1}{2}(\text{arctan}(1) - \text{arctan}(0))\). Knowing special values of the arctangent function, specifically that \(\text{arctan}(1)\) is \(\frac{\pi}{4}\) and \(\text{arctan}(0)\) is 0, we can conclude the result \(\frac{\pi}{8}\), giving us the final value of the definite integral.