Question

Use the double-angle formulas to evaluate the following integrals. \(\int \sin ^{2} x \cos ^{2} x d x\)

Short answer

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

Step-by-step solution

Recognize the Task

We need to evaluate the integral \( \int \sin^2 x \cos^2 x \, dx \) using trigonometric identities, specifically the double-angle formulas.

Apply Double-Angle Trigonometric Identity

Identify the double-angle identity for cosine: \( \cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x \). Also, recognize that \( \sin^2 x \cos^2 x = \frac{1}{4}(2\sin x \cos x)^2 \).

Use the Sine Double-Angle Identity

Rewrite \( 2\sin x \cos x = \sin 2x \), thus \( (2\sin x \cos x)^2 = \sin^2 2x \). This means \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2 2x \).

Rewrite the Integral

Substitute \( \sin^2 x \cos^2 x \) with \( \frac{1}{4} \sin^2 2x \), leading to \( \int \sin^2 x \cos^2 x \, dx = \int \frac{1}{4} \sin^2 2x \, dx \).

Simplify Using a Power-Reducing Identity

Use the identity \( \sin^2 2x = \frac{1 - \cos 4x}{2} \). Thus, \( \int \frac{1}{4} \sin^2 2x \, dx = \int \frac{1}{8}(1 - \cos 4x) \, dx \).

Integrate the Expression

Integrate each component separately: \( \int \frac{1}{8}(1 - \cos 4x) \, dx = \frac{1}{8} \int 1 \, dx - \frac{1}{8} \int \cos 4x \, dx \). \( \frac{1}{8} \int 1 \, dx = \frac{1}{8}x \) and \( \frac{1}{8} \int \cos 4x \, dx = \frac{1}{32} \sin 4x \).

Write the Final Solution

Combine the results from the integral: \[ \frac{1}{8} x - \frac{1}{32} \sin 4x + C \], where \( C \) is the constant of integration.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental in the analysis and computation of trigonometric functions. They provide relationships between trigonometric functions, allowing us to simplify complex expressions. When dealing with integrals such as \( \int \sin^2 x \cos^2 x \, dx \), identities such as those involving double angles become particularly useful.
The double-angle formulas are one of the many sets of identities that we use to transform trigonometric expressions. For example:

• \( \cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x \)
• \( \sin 2x = 2\sin x \cos x \)

In the context of our integral, recognizing that the expression \( \sin^2 x \cos^2 x \) can be rewritten using these identities greatly simplifies the process. By converting \( \sin^2 x \cos^2 x = \frac{1}{4}(2\sin x \cos x)^2 = \frac{1}{4}\sin^2 2x \), we can manage the integration in a more straightforward way.

Power-Reducing Identities

Power-reducing identities are another group of trigonometric identities that are often used to simplify the integration of functions involving powers of sine or cosine. These identities express higher powers of trigonometric functions in terms of the first power or zero power. This is extremely helpful in integration processes.
For instance, in our exercise, the power-reducing identity for sine helps in further manipulation of \( \sin^2 2x \):

• \( \sin^2 2x = \frac{1 - \cos 4x}{2} \)

By transforming \( \sin^2 2x \) into a form that involves a function of the first power, it becomes easier to split the integral into separately manageable terms. Specifically, this transformation allows us to write \( \int \frac{1}{4} \sin^2 2x \, dx \) as \( \int \frac{1}{8}(1 - \cos 4x) \, dx \), thus reducing the complexity of the integration.

Integration Techniques

Integration techniques often involve strategies that make complex integrals more tractable by transforming them into simpler, equivalent forms. When dealing with trigonometric functions, identities play a critical role in this process.
Once the expression in our problem was simplified using double-angle and power-reducing identities, we could proceed to integrate with ease. The integral \( \int \frac{1}{8}(1 - \cos 4x) \, dx \) is decomposed into two simpler integrals:\( \int \frac{1}{8} \, dx \) and \( -\int \frac{1}{8}\cos 4x \, dx \).
The integration process then involves direct application of basic integral formulas:

• \( \int 1 \, dx = x \)
• \( \int \cos 4x \, dx = \frac{1}{4}\sin 4x \)

By completing these integrals, we obtain \( \frac{1}{8} x - \frac{1}{32} \sin 4x + C \), which represents the solution to the original problem. Integrating trigonometric expressions can include a combination of techniques like substitution, using identities, and decomposing expressions, which helps in deriving concise solutions.