Problem 67 - Calculus Single Variable

Question

When \(\sin (x)\) and \(\cos (x)\) are both raised to the same positive power in an integrand, the identity \(\sin (2 x)=2 \sin (x) \cos (x)\) may be used to simplify the integral. Use this observation as the basis for calculation of the integrals. $$ \int \cos ^{2}(x) \sin ^{2}(x) d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{x}{8} - \frac{1}{32} \sin(4x) + C \).

Step 1: Identify Trigonometric Identity Usage

We notice that both \( \sin(x) \) and \( \cos(x) \) are raised to the second power in the integral. To simplify, we can use the double angle identity for sine: \[ \sin(2x) = 2 \sin(x) \cos(x). \] This identity suggests a potential change to our integral.

Step 2: Rewriting Using the Identity

Starting from \( \sin(2x) = 2 \sin(x) \cos(x) \), we find \( \sin(x) \cos(x) = \frac{1}{2} \sin(2x) \). Rewrite the integral as follows: \( \cos^2(x) \sin^2(x) = \left( \frac{1}{2} \sin(2x) \right)^2 \).

Step 3: Simplify the Expression

We simplify \( \left( \frac{1}{2} \sin(2x) \right)^2 \) to \( \frac{1}{4} \sin^2(2x) \). So our integral becomes \( \int \frac{1}{4} \sin^2(2x) \, dx \). Taking out the constant factor, we have \( \frac{1}{4} \int \sin^2(2x) \, dx \).

Step 4: Use the Reduction Formula for \( \sin^2(2x) \)

We use the trigonometric identity \( \sin^2(A) = \frac{1 - \cos(2A)}{2} \) to reduce the integral. Substitute \( 2x \) for \( A \), \( \sin^2(2x) = \frac{1 - \cos(4x)}{2} \). Thus, the integral becomes \( \frac{1}{4} \int \frac{1 - \cos(4x)}{2} \, dx \).

Step 5: Simplify the Integral

This can be further simplified to \( \frac{1}{8} \int (1 - \cos(4x)) \, dx \), and split into two integrals: \( \frac{1}{8} \left( \int 1 \, dx - \int \cos(4x) \, dx \right) \).

Step 6: Compute Each Integral

Compute the integral of each part: \( \int 1 \, dx = x \), and \( \int \cos(4x) \, dx = \frac{1}{4} \sin(4x) \). Thus, the expression becomes \( \frac{1}{8} \left( x - \frac{1}{4} \sin(4x) \right) \).

Step 7: Final Simplification

Combine the terms to get \( \frac{x}{8} - \frac{1}{32} \sin(4x) + C \), where \( C \) is the constant of integration. This is the most simplified form of the integral.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in calculus, helping to simplify complex expressions involving trigonometric functions. An identity like \( \sin(2x) = 2 \sin(x) \cos(x) \) shows the relationship between the sine of a double angle and the product of sine and cosine of a single angle. These identities make it possible to transform expressions so they are easier to integrate or differentiate.

• Some common trigonometric identities include Pythagorean identities, addition and subtraction formulas, and double angle formulas.
• In our case, the identity \( \sin(2x) = 2 \sin(x) \cos(x) \) is used to substitute and transform \( \sin^2(x) \cos^2(x) \) into a format conducive to further calculations.

This approach often reduces the level of complexity, allowing us to handle integrals more conveniently by transforming them into integrals of single trigonometric functions or through polynomial terms.

Double Angle Formulas

Double angle formulas are a special subset of trigonometric identities. They provide a connection between the functions of an angle and its double, and are very useful in calculating integrals and derivatives. The formula \( \sin(2x) = 2 \sin(x) \cos(x) \) is crucial when dealing with integrals involving products of powers of sine and cosine.

• These formulas simplify expressions by reducing the number of different trigonometric terms involved.
• Using a double angle formula allows converting a product of trigonometric functions into a single trigonometric function with a double angle, as seen in our integral \( \int \cos^2(x) \sin^2(x) \,dx \).

Double angle formulas are extremely helpful in simplifying solutions, especially when dealing with complicated trigonometric integrals.

Reduction Formulas

Reduction formulas are techniques that transform integrals into simpler forms, often exploiting trigonometric identities and algebraic manipulations. They are used to simplify powers of trigonometric functions, as shown in the integral \( \int \sin^2(2x) \, dx \).

• By using the identity \( \sin^2(A) = \frac{1 - \cos(2A)}{2} \), we express the integral in terms of a simpler expression using cosine.
• This helps in breaking down a complex integral into more manageable pieces for evaluation, such as turning an integral of \( \sin^2(2x) \) into separate integrals involving constant terms and cosine functions.

The advantage of using reduction formulas is that they lead to integrals that can be solved using basic integration rules, paving the way for solutions that are easier to interpret and apply.

Calculus

Calculus is fundamentally about the study of change and motion, and integration is a key component of this analysis. In this integration problem, calculus involves transforming the integrand using relevant identities and solving the resulting expressions. The whole process involves breaking down a challenging integral into simpler parts which can be easily evaluated.

• Integration, particularly of trigonometric functions, often requires recognizing patterns and applying identities like those of double angles or reductions.
• By understanding the calculus involved in integrating \( \int \cos^2(x) \sin^2(x) \, dx \), students can appreciate how transformations and identity applications make complex calculus problems manageable.

Mastery of calculus concepts, including trigonometric integrals, gives students powerful tools to tackle a wide range of problems, enhancing their problem-solving skills and mathematical creativity.