91Ó°ÊÓ

Integration by Substitution is a powerful technique used to evaluate more complex integrals, especially when direct integration is challenging. In the provided exercise, we start by evaluating the inner integral \( \int_{1}^{2} x \cos(xy) \, dy \). Here, substitution helps simplify the integral by changing the variable to easily integrable terms.

To apply substitution, let \( u = xy \), which implies that the differential \( du = x \, dy \) or \( dy = \frac{du}{x} \). This substitution shows us how the constants and limits transform. When \( y = 1 \), \( u = x \); and when \( y = 2 \), \( u = 2x \). Thus, the integral becomes \( \int_{x}^{2x} \cos(u) \, du \). This is a clearer format for easily integrating trigonometric functions.

After conversion, one can directly evaluate the integral using basic trigonometric antiderivatives, ensuring that the complex function becomes much more manageable.

Definite Integration involves calculating the integral with specific upper and lower limits, effectively computing the net area under a curve within these bounds. In this exercise, after transforming the inner integral, definite integration comes into play.

Specifically, when evaluating the integral \( \int_{x}^{2x} \cos(u) \, du \), we use the antiderivative of \( \cos(u) \), which is \( \sin(u) \). The definite integral is calculated as \( \sin(2x) - \sin(x) \), providing an expression ready for the next step: the outer integration.

Finally, the outer integral involves further definite integration, \( \int_{\pi/2}^{\pi} \left[ \sin(2x) - \sin(x) \right] \ dx \). Each function in the expression is integrated separately: \( \sin(2x) \) becomes \(-\frac{1}{2} \cos(2x)\), and \( \sin(x) \) becomes \(-\cos(x)\). The evaluated limits \( [\pi/2, \pi] \) yield the final result. This two-step integration process ensures precision and thoroughness in calculating the area or value represented by the integral.

Trigonometric Functions like \( \sin \) and \( \cos \) are fundamental in calculus, especially when dealing with periodic patterns and oscillations. In this exercise, trigonometric identities and functions play an essential role in simplifying and computing the integrals.

The problem involves functions of the form \( x \cos( xy ) \) and later \( \sin \) and \( \cos \) as part of the integrals. During substitution, the integral \( \int \cos(u) \, du \) uses the direct antiderivative \( \sin(u) \). Understanding these antiderivatives helps in finding the areas under sine and cosine curves.

Additionally, trigonometric values at key angle measures such as \( \pi \), \( \pi/2 \), and \( 2\pi \) simplify the evaluation process, where \( \cos(2\pi) = 1 \), \( \cos(\pi) = -1 \), and \( \cos(\pi/2) = 0 \).

These fundamental concepts show the intertwining of calculus with trigonometry, simplifying complex calculations into routine mathematical processes.