Question

Identify each of the following integrals or expressions as one of the functions of this chapter. Check your work by evaluating both your answer and the original problem by computer. Be sure you understand your computer program's notation. $$\int_{0}^{\pi / 2} \sqrt{\sin ^{3} \theta \cos ^{5} \theta} d \theta$$

Short answer

The integral equals \( \frac{\Gamma\left( \frac{5}{4} \right) \Gamma\left( \frac{7}{4} \right)}{4} \).

Step-by-step solution

Simplify the integrand

Rewrite the integrand \( \sqrt{\sin^3 \theta \cos^5 \theta} \) using the property of exponents inside the square root. The expression can be simplified to \( \sqrt{\sin^3 \theta \cos^5 \theta} = (\sin^3 \theta \cos^5 \theta)^{1/2} = \sin^{3/2} \theta \cos^{5/2} \theta \)

Identify the standard integral

Recognize the standard form for the integral of the type \( \int_0^{\pi / 2} \sin^m \theta \cos^n \theta \, d\theta \). In this case, \( m = 3/2 \) and \( n = 5/2 \). The formula for this type of integral is given by: \ \[ \int_0^{\pi / 2} \sin^m \theta \cos^n \theta \, d\theta = \frac{1}{2} B\left(\frac{m+1}{2}, \frac{n+1}{2}\right) \] \ where \( B(x,y) \) denotes the Beta function.

Apply the Beta function

Apply the formula: \ \[ B\left(\frac{3/2+1}{2}, \frac{5/2+1}{2}\right) = B\left(\frac{5/2}{2}, \frac{7/2}{2}\right) = B\left(\frac{5}{4}, \frac{7}{4}\right) \] \ The Beta function \( B(x, y) \) can be computed using: \ \[ B\left(x, y\right) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} \]

Evaluate using Gamma function properties

The Gamma function \( \Gamma(n) \) has properties such as \( \Gamma(n) = (n-1)! \) for positive integers. However, when dealing with fractions: \ \[ \Gamma\left(\frac{5}{4}\right) \text{ and } \Gamma\left(\frac{7}{4}\right) \] need more advanced properties or computational tools to evaluate directly. For the given integral, the value can be obtained from a known result: \ \[ \int_0^{\pi/2} \sin^{3/2} \theta \cos^{5/2} \theta \, d\theta = \frac{\Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}{2\Gamma\left(3\right)} \]

Final evaluation

Given \( \Gamma(3) = 2! = 2 \), the integral becomes: \ \[ \int_0^{\pi / 2} \sin^{3/2} \theta \cos^{5/2} \theta \, d\theta = \frac{ \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}{2 \cdot 2} = \frac{ \Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{7}{4}\right)}{4} \]

Key concepts

Beta function

The Beta function, denoted as \ B(x, y) \, is a special function that plays a crucial role in calculus and analysis. It's defined for two positive real numbers x and y and is related to integration involving powers of sine and cosine. The integral representation of the Beta function is given by: \[ B(x, y) = \int_0^{1} t^{x-1} (1-t)^{y-1} dt \] This function ties together products of Gamma functions, often helping to solve integrals that involve these forms. When dealing with certain trigonometric integrals, recognizing a pattern that matches the Beta function can immensely simplify your work.

Gamma function

The Gamma function, \( \ \Gamma(n) \ \), extends the factorial function to real and complex numbers. While the factorial function, denoted n!, is defined for natural numbers, the Gamma function is more versatile. For a positive integer n, it holds that: \[ \Gamma(n) = (n-1)! \] However, the Gamma function smoothly extends to non-integer values as well. For example, evaluating \( \ \Gamma(5/4) \ \) or \( \ \Gamma(7/4) \ \) requires more advanced methods or numerical computation. Understanding the properties of the Gamma function, such as \( \ \Gamma(x+1) = x \ \Gamma(x) \ \), is invaluable when dealing with complex integrals that involve these extended factorials.

Integral evaluation

Integral evaluation often requires transforming complex expressions into known forms. In the step-by-step solution given, the integrand \( \sqrt{\text{sin}^3 \theta \ \text{cos}^5 \theta} \) was simplified to \( \text{sin}^{3/2} \theta \ \text{cos}^{5/2} \theta \). Recognizing this pattern allowed us to apply the standard integral formula involving the Beta function: \[ \int_0^{\pi/2} \text{sin}^m \theta \ \text{cos}^n \theta \ d \theta = \frac{1}{2} \ B \left(\frac{m+1}{2}, \frac{n+1}{2}\right) \ \] By correctly identifying this form, we can utilize special functions instead of solving it in more cumbersome ways. This integral involved gamma functions of fractions, providing an example of how integral evaluation connects with special functions like Beta and Gamma.

Trigonometric integrals

Trigonometric integrals involving sine and cosine functions often appear in calculus. They can be evaluated using various methods, including substitution, symmetry, and recognizing patterns for special functions. A common integral form is: \[ \int_{0}^{\pi / 2} \sin^m \theta \cos^n \theta \ d\theta \] For this type, Beta and Gamma functions frequently offer a solution. Simplifying the integrand by recognizing powers as seen in the given example, leads to applying the Beta function. For instance, the integral \( \int_{0}^{\pi / 2} \sin^{3/2} \theta \cos^{5/2} \theta \ d\theta \) straightforwardly transforms to: \[ \frac{\Gamma \left(\frac{5}{4}\right) \Gamma \left(\frac{7}{4}\right)}{4}\] Trigonometric integrals not only require familiarity with trigonometric identities but also with advanced functions for efficient solving.