Problem 13 - Essential Calculus

Question

Find the integral. $$ \int \sin ^{2} \alpha \cos ^{2} \alpha d \alpha $$

Step-by-Step Solution

Answer

The integral of \(\int \sin ^{2} \alpha \cos ^{2} \alpha d \alpha\) is \(\frac{1}{8}\alpha - \frac{1}{32} \sin(4 \alpha) + C.\)

Step 1: Identity Substitution

Identify and utilise the double-angle identity which states that \(\sin^{2}x \cos^{2}x = \frac{1}{4}\sin ^2(2x)\). Rewrite the integral as \(\frac{1}{4}\int \sin ^2 (2 \alpha) d \alpha.\)

Step 2: Recurrence Sinusoidal Identity

Use the recurrence sinusoidal identity \(\sin ^2 (x) = \frac{1}{2} (1- \cos(2x))\). Substituting \(\sin ^2 (2 \alpha)\) with \(\frac{1}{2} (1- \cos(4 \alpha))\), the integral now reads \(\frac{1}{8}\int (1- \cos(4 \alpha)) d \alpha.\)

Step 3: Separate the Integrals

Separate the integral into two parts: \(\frac{1}{8} \int d \alpha - \frac{1}{8} \int \cos(4 \alpha) d \alpha.\)

Step 4: Integrate

Integrate each part separately. The integral of \(1\) with respect to \(\alpha\) is \(\alpha\), and the integral of \(\cos(4 \alpha)\) with respect to \(\alpha\) is \(\frac{1}{4}\sin(4 \alpha)\). So, the integral becomes \(\frac{1}{8}\alpha - \frac{1}{32} \sin(4 \alpha) + C\), where \(C\) represents the constant of integration.

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve expressions that include trigonometric functions such as sine, cosine, tangent, and their multiples. These integrals are critical in calculus as they appear in various applications, including physics and engineering.

• When dealing with integrals of trigonometric functions, various identities can simplify the process. For instance, integrating \( \int \sin^n(x) \cos^m(x) dx \), where \( n \) and \( m \) are integers, often involves using identities to simplify the powers or arguments.
• In our original exercise, we had to integrate \( \sin^2(\alpha)\cos^2(\alpha) \). Simplifying such expressions usually involves leveraging known trigonometric identities.

Understanding trigonometric integrals is essential for tackling complex calculus problems involving waves, oscillations, and other periodic phenomena. It often requires a strong grasp of trigonometric identities and their applications.

Double-Angle Identity

The double-angle identities are formulas that express trigonometric functions of double angles \((2x)\) in terms of single angles \((x)\). These identities are particularly helpful for simplifying expressions during integration.

• For sine and cosine, these identities are given by:
  \[ \sin(2x) = 2\sin(x)\cos(x) \] \[ \cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x) \]
• In the context of integrals, such as in our example, the double-angle identity for sine was used to transform \( \sin^2(\alpha)\cos^2(\alpha) \) into a simpler form:
  \( \sin^2(\alpha)\cos^2(\alpha) = \frac{1}{4}\sin^2(2\alpha) \).

Understanding and applying the double-angle identities allows for the simplification of integrals that otherwise appear complex, making integration more manageable.

Recurrence Sinusoidal Identity

The recurrence sinusoidal identity relates to the way trigonometric squares can be expressed in terms of double angles. The identity is helpful in managing and simplifying trigonometric integrals.

• The key identity used is:
  \[ \sin^2(x) = \frac{1}{2} (1- \cos(2x)) \] This equation converts a sine squared term into an expression involving a cosine term, which is often easier to integrate.
• In the given exercise, the recurrence sinusoidal identity was crucial
  for transforming \( \sin^2(2\alpha) \) to \( \frac{1}{2} (1- \cos(4\alpha)) \), making the integration process considerably simpler.

These identities are foundational in calculus, providing a method to break down complex trigonometric integrals into more straightforward components that are easier to integrate directly.

Definite and Indefinite Integrals

Integrals can be either definite or indefinite. Understanding the difference is essential for solving calculus problems.

• An **indefinite integral** is represented as \( \int f(x) \, dx \) and results in a family of functions,
  including an arbitrary constant \(C\), as it represents the antiderivative.
• A **definite integral** has limits \([a, b]\), written as \( \int_{a}^{b} f(x) \, dx \), producing a number that represents the net area under the curve \( f(x) \) from \( a \) to \( b \).
• In our exercise, we dealt with an indefinite integral, resulting in:
  \[ \frac{1}{8} \alpha - \frac{1}{32} \sin(4\alpha) + C \] where \(C\) is the constant of integration.

Recognizing when to apply definite versus indefinite integrals is critical. It influences the interpretation of the integral in any applied context such as physics, where definite integrals often quantify physical spaces or changes, and indefinite integrals model general functions over time or space.