91Ó°ÊÓ

Integration is a fundamental concept in calculus, focusing on finding the area under curves. It's like adding up small parts to find the whole. In the exercise, we're dealing with a definite integral from \(-\pi / 4\) to 0 with the integrand \(\frac{\sin x}{\cos^2 x}\). The process of integration involves working with functions to accumulate quantities. There are various techniques to perform integration, and in this case, a trigonometric substitution is used to simplify the expression. Trigonometric identities and properties help ease the integration process for complex expressions. For instance, here, we substitute \(u = \cos x\), which transforms our integral into a simpler form that can be integrated easily.

The change of variables, also known as substitution, is a powerful method used in calculus to simplify complex integrals. In our exercise, we apply a trigonometric substitution to change variables, from \(x\) to \(u\). By letting \(u = \cos x\), we simplify our integrand. The differential \(du = -\sin x \, dx\) lets us substitute and transform this integral into a more straightforward expression in terms of \(u\):- Start with the original integrand \(\frac{\sin x}{\cos^2 x}\)- Substitute \(\sin x \, dx = -du\) to modify the integral to \(\frac{-1}{u^2}\)\.This transformation makes the integration manageable and allows us to proceed to calculate the indefinite integral. The change of variables often helps to simplify the components, aligning them in a more straightforward form.

Definite integrals extend the concept of indefinite integrals by incorporating limits of integration, which correspond to the bounds of the area we are calculating. Here, we compute \(\int_{-\pi / 4}^{0} \frac{\sin x}{\cos^2 x} \, dx\) by evaluating the difference between the antiderivative at the upper and lower limits. After performing the substitution and integrating, we return to the original variable \(x\), resulting in \(\frac{1}{\cos x}\). We then evaluate from \(-\pi / 4\) to 0:- Substitute the upper limit \(\cos(0) = 1\)- Substitute the lower limit \(\cos(-\pi/4) = \frac{\sqrt{2}}{2}\)- Final computation: \(1 - \frac{\sqrt{2}}{2}\)Definite integrals provide a specific value, representing the total area, or integral, over a given interval.