Question

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

Short answer

\(\frac{1}{8}x - \frac{1}{32}\sin(4x) + C\)

Step-by-step solution

Use the double-angle identity

We begin by using the trigonometric identity for the double angle. One useful identity here is:\[\sin^2 x \cos^2 x = \left(\frac{\sin(2x)}{2}\right)^2 = \frac{\sin^2(2x)}{4}\]This transforms our integral into an expression that involves the sine of a double angle, which might be easier to integrate.

Substitute and simplify the integral

Using the identity from Step 1, the integral becomes:\[\int \sin^2 x \cos^2 x \, dx = \int \frac{\sin^2(2x)}{4} \, dx = \frac{1}{4} \int \sin^2(2x) \, dx\]Now, we have simplified the integral by expressing it in terms of \(\sin^2(2x)\).

Apply the power-reducing formula

To further simplify \(\sin^2(2x)\), we apply the power-reducing formula:\[\sin^2(2x) = \frac{1 - \cos(4x)}{2}\]Therefore, the integral becomes:\[\frac{1}{4} \int \sin^2(2x) \, dx = \frac{1}{4} \int \frac{1 - \cos(4x)}{2} \, dx = \frac{1}{8} \int (1 - \cos(4x)) \, dx\]

Integrate the simplified expression

Now, split the integral and integrate term by term:\[\frac{1}{8} \int (1 - \cos(4x)) \, dx = \frac{1}{8} \left( \int 1 \, dx - \int \cos(4x) \, dx \right)\]The integrals are:\[\int 1 \, dx = x\]\[\int \cos(4x) \, dx = \frac{1}{4}\sin(4x)\]So, combining these results gives us:\[\frac{1}{8} \left( x - \frac{1}{4} \sin(4x) \right) + C\]

Simplify the final expression

Simplify the solution by distributing the constant inside:\[\frac{1}{8}x - \frac{1}{32}\sin(4x) + C\]This is the final simplified form of the integral.

Key concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between various trigonometric functions. These identities are crucial in simplifying expressions and solving integrals that involve trigonometric functions. In the context of this problem, we primarily use a special trigonometric identity known as the double-angle identity. This identity allows us to express the product of \( \sin^2 x \cos^2 x \) in terms of a single trigonometric function of a doubled angle:- The identity used here is: \( \sin^2 x \cos^2 x = \left( \frac{\sin(2x)}{2} \right)^2 \), which simplifies to \( \frac{\sin^2(2x)}{4} \).By using these identities, we convert the original integral into a more manageable form. It is a powerful technique that helps streamline the process of integration by reducing complex expressions to simpler forms. Understanding and practicing these identities will improve your ability to manipulate and solve trigonometric integrals effectively.

Integrals

An integral is a fundamental concept in calculus used to calculate the area under a curve. In this exercise, we are asked to integrate the function \( \sin^2 x \cos^2 x \). Solving integrals, especially those involving trigonometric functions, can often be challenging. However, by transforming trigonometric expressions using identities such as the double-angle or power-reducing formulas, we can make the integration process more straightforward.Here's how the integral transformation and solution unfolds in our problem:- Start with the expression \( \sin^2 x \cos^2 x \) and use the double-angle formula to rewrite it as \( \frac{\sin^2(2x)}{4} \).- Rewrite the integral in terms of this new expression and simplify it:\[ \int \sin^2 x \cos^2 x \, dx = \frac{1}{4} \int \sin^2(2x) \, dx \]- Next, use further transformations such as the power-reducing formula to integrate.This process demonstrates how integrals combined with trigonometric identities can be maneuvered to reach a solution efficiently. Knowing when and how to apply these transformations is critical in calculus.

Power-Reducing Formulas

Power-reducing formulas are specific trigonometric identities that simplify powers of sine and cosine to expressions involving the first power of a cosine. These formulas are very useful in calculus, particularly for integrating functions like our example here, \( \sin^2(2x) \).The power-reducing formula we use is:\[ \sin^2(2x) = \frac{1 - \cos(4x)}{2} \]Applying this formula, allows us to further simplify the integrand from \( \sin^2(2x) \) to a more straightforward expression \( \frac{1 - \cos(4x)}{2} \). This is crucial because:- It breaks down a complex trigonometric power into manageable linear terms.- We divide the integral into parts: \( \int \frac{1}{2} \left( 1 - \cos(4x) \right) \, dx \), making the integral easier to solve.- The integration of \(1\) and \(\cos(4x)\) becomes direct, resulting in \( x \) and \( \frac{1}{4} \sin(4x) \), respectively.Therefore, understanding power-reducing formulas is essential for tackling integrals involving squared trigonometric functions. These formulas provide a systematic way to simplify expressions and find solutions effectively in calculus.