Question

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

Short answer

The integral evaluates to \( \frac{x}{4} + \frac{1}{16} \sin(4x) - \frac{1}{8} \sin(2x) - \frac{1}{48} \sin(6x) + C \).

Step-by-step solution

Apply Double-Angle Formulas

The first step in solving this integral is to use the double-angle formulas for sine and cosine to simplify the integrand. The formulas are: \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) and \( \cos^2(2x) = \frac{1 + \cos(4x)}{2} \). Rewrite the integrand \( \sin^2(x) \cos^2(2x) \) using these formulas, resulting in:\[\left(\frac{1 - \cos(2x)}{2}\right) \left(\frac{1 + \cos(4x)}{2}\right) = \frac{(1 - \cos(2x))(1 + \cos(4x))}{4}.\]

Simplify the Expression Inside the Integral

Next, expand the expression inside the integral: \((1 - \cos(2x))(1 + \cos(4x))\). This results in:\[1 \cdot 1 + 1 \cdot \cos(4x) - \cos(2x) \cdot 1 - \cos(2x) \cdot \cos(4x) = 1 + \cos(4x) - \cos(2x) - \cos(2x)\cos(4x).\]So the integrand becomes \( \frac{1}{4}(1 + \cos(4x) - \cos(2x) - \cos(2x)\cos(4x)) \).

Simplify and Split the Integral

Integrate each term separately:\[\int \left(\frac{1}{4}\right)(1 + \cos(4x) - \cos(2x) - \cos(2x)\cos(4x)) \, dx = \frac{1}{4} \int (1) \, dx + \frac{1}{4} \int \cos(4x) \, dx - \frac{1}{4} \int \cos(2x) \, dx - \frac{1}{4} \int \cos(2x)\cos(4x) \, dx.\]

Compute the Simpler Integrals

The basic integrals can be computed as follows:1. \( \int 1 \, dx = x. \)2. \( \int \cos(4x) \, dx = \frac{1}{4} \sin(4x). \)3. \( \int \cos(2x) \, dx = \frac{1}{2} \sin(2x). \)

Solve the Product Integral Using Product-to-Sum Identities

The integrand \( \cos(2x)\cos(4x) \) can be rewritten using the product-to-sum identities:\[\cos(A)\cos(B) = \frac{1}{2} [\cos(A - B) + \cos(A + B)].\]Applying this gives \( \cos(2x)\cos(4x) = \frac{1}{2}[\cos(2x + 4x) + \cos(2x - 4x)] = \frac{1}{2}[\cos(6x) + \cos(-2x)] \). Substitute back and integrate each term separately. This results in:\[\frac{1}{2} \int [\cos(6x) + \cos(2x)] \, dx = \frac{1}{2} \left( \frac{1}{6} \sin(6x) \right) + \frac{1}{2} \left( \frac{1}{2} \sin(2x) \right).\]

Combine All Parts and Simplify

After computing each section of the integral, combine all results:\[\frac{1}{4} \left( x + \frac{1}{4} \sin(4x) - \frac{1}{2} \sin(2x) - \left( \frac{1}{12} \sin(6x) + \frac{1}{4} \sin(2x) \right) \right) + C.\]Simplify by combining like terms. The final answer, after simplification, is:\[\frac{x}{4} + \frac{1}{16} \sin(4x) - \frac{1}{8} \sin(2x) - \frac{1}{48} \sin(6x) + C.\]

Conclusion: Final Result

After completing all steps and simplifications, the evaluated integral is \( \frac{x}{4} + \frac{1}{16} \sin(4x) - \frac{1}{8} \sin(2x) - \frac{1}{48} \sin(6x) + C \).

Key concepts

Trigonometric Integration

Trigonometric integration is a technique used to solve integrals involving trigonometric functions like sine and cosine. This often requires manipulation and simplification using various trigonometric identities. In the exercise given, double-angle formulas are employed to simplify the integrand before integration.

Double-angle formulas are key in trigonometric integration as they help to reduce complex expressions into simpler forms. For example, we use the following:

• \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) which relates a square of sine function to cosine of a double angle.
• \( \cos^2(2x) = \frac{1 + \cos(4x)}{2} \) which connects a square of cosine function to cosine of four times the angle.

This simplification is crucial as it reduces the complexity of handling the integrals individually.

Product-to-Sum Identities

In trigonometric calculus, product-to-sum identities transform the product of trigonometric functions into a sum or difference of similar functions, which are easier to integrate. Consider the identity:

• \( \cos(A)\cos(B) = \frac{1}{2} [\cos(A + B) + \cos(A - B)] \).

This transformation aids in simplifying terms to a form where basic integration can occur more smoothly.

In the exercise, we encountered the product \(\cos(2x)\cos(4x)\). By applying the product-to-sum formula, this product was transformed into:

• \( \cos(6x) + \cos(2x) \).

By breaking down complex products into simpler sums, integration of each term becomes more straightforward and possible with basic integration techniques.

Integration Techniques

Integration techniques with trigonometric functions often focus on breaking down the integrals into simpler parts and using identities like double-angle and product-to-sum.

The key steps in solving such integrals include:

• First, simplification of the integrand using trigonometric identities like double angles to make the expression less complex.
• Next, decomposition of products using product-to-sum identities to enable easier integration of each resulting term.
• Finally, solving the basic integrals using standard methods: \( \int \cos(kx) \ dx = \frac{1}{k} \sin(kx) \) and replacing the values in the original integrals to obtain the final result.

These structured approaches ensure the integration process is efficient, avoiding complications from direct integration of mixed trigonometric products.