91Ó°ÊÓ

Trigonometric substitution is a technique used in integration to simplify integrals that contain square roots of quadratic expressions. It is especially useful when the integral involves expressions like \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\), or \(\sqrt{x^2 - a^2}\). The key idea is to use trigonometric identities to replace these expressions. This often helps in converting the integral into a simpler form that is easier to solve.

Here’s how trigonometric substitution works in practice:

• For \( \sqrt{a^2 - x^2} \), use the substitution \( x = a \sin(\theta) \), which transforms \( dx = a \cos(\theta) \, d\theta \).
• For \( \sqrt{a^2 + x^2} \), use \( x = a \tan(\theta) \), resulting in \( dx = a \sec^2(\theta) \, d\theta \).
• For \( \sqrt{x^2 - a^2} \), use \( x = a \sec(\theta) \) with \( dx = a \sec(\theta) \tan(\theta) \, d\theta \).

In the given exercise, we encounter an integral with \( \sqrt{1-y^2} \) in the denominator. Here, the substitution \( y = \sin(\theta) \) is used. This is a beneficial choice because the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) directly helps in simplifying the expression. Thus, the integral \( \int_{0}^{\sqrt{3} / 2} \frac{d y}{\left(1-y^{2}\right)^{5 / 2}} \) gets transformed into a function of \( \theta \), which is simpler to integrate.

Reduction formulas are powerful tools in the integration process, particularly for handling integrals raised to a power or involving trigonometric functions. They help break down a complex integral into a simpler form by expressing it in terms of an integral with a lower power.

A reduction formula often involves recursion, allowing an integral of one form to be reduced iteratively. For example, consider the integral \( \int \sec^n(\theta) \, d\theta \). A reduction formula for this is:

• \( \int \sec^n(\theta) \, d\theta = \frac{1}{n-1} \sec^{n-2}(\theta)\tan(\theta) + \frac{n-2}{n-1} \int \sec^{n-2}(\theta) \, d\theta \).

In our solution, the reduction formula for \( \int \sec^4(\theta) \, d\theta \) is particularly useful. It allows us to express this integral in terms of \( \int \sec^2(\theta) \, d\theta \), which is simpler and readily known since \( \int \sec^2(\theta) \, d\theta = \tan(\theta) \). This method of reduction is instrumental in solving integrals without directly computing heavy algebraic expressions.

Definite integrals are used to compute the total accumulation of a quantity, which can represent area under a curve, among other applications. They are denoted by \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. Unlike indefinite integrals, which provide a family of functions, definite integrals calculate a specific numerical value.

When working with definite integrals:

• First, ensure the integral is properly set up within the given bounds. Use substitution when necessary to simplify the function within the integration limits.
• With trigonometric substitution, remember to adjust the limits of integration to reflect the new variable. This often involves evaluating the resulting functions at the new bounds.
• After evaluating the integral, use these bounds to obtain the result by finding the difference \( F(b) - F(a) \).

In our problem, we began with the limits \( 0 \) to \( \sqrt{3}/2 \) in terms of \( y \). Once the trigonometric substitution was applied, these were transformed to \( 0 \) to \( \pi/3 \) in terms of \( \theta \). Calculating the definite integral over this range allowed the final numerical result \( \frac{10\sqrt{3}}{9} \) to be obtained.