91Ó°ÊÓ

The substitution method is a powerful integration technique used to simplify complex integrals, especially those involving composite functions. The goal is to transform the original integral into a more manageable form by introducing a new variable. This is often useful when your original expression includes cumbersome and convoluted terms, like nested functions or complex expressions under radicals.

In our example, the integral \( \int(x \cos x + \sin x) \sqrt{1 + x^{2} \sin^{2} x} \, dx \) was initially quite complex due to the square root containing the expression \( x^2 \sin^2 x \). By recognizing \( u = x \sin x \), we simplify the expression inside the square root to \( \sqrt{1 + u^2} \), making the integral much easier to evaluate.

When applying substitution:

• Choose a substitution \( u = g(x) \) that simplifies the integral expression.
• Differentiate \( u \) with respect to \( x \) to get \( du = g'(x) \, dx \).
• Replace all \( x \)-terms and \( dx \) in the integral with \( u \) and \( du \).
• Evaluate the new integral in terms of \( u \).
• Re-substitute to express the final solution in terms of \( x \).

The product rule is a cornerstone differentiation technique used when taking derivatives of products of two functions. It is essential for correctly handling integrals that involve products when creating substitutions.

The product rule formula is given by:
\[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\]
This rule was applied in the step where we differentiated \( u = x \sin x \), resulting in:
\[du = (x \cos x + \sin x) \, dx\]
This corresponds to assigning \( f(x) = x \) and \( g(x) = \sin x \), thus making \( f'(x) = 1 \) and \( g'(x) = \cos x \). Substituting these into the product rule yields \( du \), which matched part of our original integrand.

Understanding the product rule is crucial because it allows us to successfully manipulate and simplify products of functions, especially when preparing for substitution during integration.

Hyperbolic functions are analogs of trigonometric functions but for the hyperbola, similar in form yet different in properties. They play an important role in some integration scenarios, particularly when dealing with integrals like \( \int \sqrt{1 + u^2} \, du \).

The hyperbolic sine and cosine functions, denoted as \( \sinh(x) \) and \( \cosh(x) \), are defined by:
\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]
\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]
These functions are useful due to their relationships with the identities \( \cosh^2(x) - \sinh^2(x) = 1 \), which mirrors the Pythagorean identity in trigonometry.

When encountering an integral like \( \int \sqrt{1 + u^2} \, du \), hyperbolic functions often simplify the process. Specifically, this integral can be related to \( \sinh^{-1}(u) \), as many CAS or advanced integration techniques recognize it as a standard form.

Utilizing hyperbolic functions in integration allows for neat and elegant solutions to otherwise cumbersome integrals.