Question

For the following exercises, integrate using whatever method you choose.\(\int \sin ^{2}(x) \cos ^{2}(x) d x\)

Short answer

\( \int \sin^2(x) \cos^2(x) \, dx = \frac{1}{8}x - \frac{1}{32}\sin(4x) + C. \)

Step-by-step solution

Use Trigonometric Identity

To simplify the integral \( \int \sin^2(x) \cos^2(x) \, dx \), we will use the trigonometric identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) and \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). This transforms the integral into:\[ \int \left(\frac{1 - \cos(2x)}{2} \right) \left(\frac{1 + \cos(2x)}{2} \right) dx. \] Now, expand the product inside the integral.

Expand the Product

Expand the product inside the integral:\[ \int \left(\frac{1 - \cos(2x)}{2} \right) \left(\frac{1 + \cos(2x)}{2} \right) dx = \int \frac{1 - \cos^2(2x)}{4} \, dx. \]Using the identity \( 1 - \cos^2(2x) = \sin^2(2x) \), the integral simplifies to:\[ \int \frac{\sin^2(2x)}{4} \, dx. \]

Use Double Angle Identity for Sine

To simplify \( \sin^2(2x) \), use the identity \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \). For our integral:\[ \sin^2(2x) = \frac{1 - \cos(4x)}{2}. \]Thus, the integral becomes:\[ \int \frac{1 - \cos(4x)}{8} \, dx. \]

Integrate Each Term

Separate the integral into two simpler integrals:\[ \int \frac{1}{8} \, dx - \int \frac{\cos(4x)}{8} \, dx. \]The first integral evaluates to:\[ \frac{1}{8}x. \]For the second integral, use substitution. Let \( u = 4x \), so \( du = 4 dx \) or \( dx = \frac{du}{4} \). Then\[ \int \frac{\cos(u)}{8} \cdot \frac{1}{4} \, du = \frac{1}{32} \int \cos(u) \, du = \frac{1}{32} \sin(4x). \]

Combine Results and Add Constant of Integration

Combine the results of the integral:\[ \frac{1}{8}x - \frac{1}{32}\sin(4x) + C. \]Thus, the integral \( \int \sin^2(x) \cos^2(x) \, dx \) evaluates to:\[ \frac{1}{8}x - \frac{1}{32}\sin(4x) + C, \]where \( C \) is the constant of integration.

Key concepts

Trigonometric Identities

Trigonometric identities are powerful tools in integral calculus, especially when dealing with sinusoidal functions like sine and cosine. These identities help simplify expressions and evaluate integrals that might otherwise be quite complex. In our exercise, we used two specific identities:

• Square Identities: \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) and \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). These allow us to rewrite the integral in terms of cosines of double angles.
• Double Angle Identity for Sine: \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \). This reduces the complexity of expressions involving squares of sine functions.

By utilizing these identities, the original integral \( \int \sin^2(x)\cos^2(x)\,dx \) is transformed into a more manageable form, allowing easier integration.

Integration Techniques

Integration techniques are strategic methods applied to evaluate integrals effectively. When direct integration is not possible, these techniques become essential. In our exercise, the integral was complicated due to the product of \( \sin^2(x) \) and \( \cos^2(x) \). By expanding the expression using the identities mentioned, it was structured into a form suitable for:

• Splitting Integrals: The expression \( \int \frac{\sin^2(2x)}{4} \, dx \) was split into simpler integrals of constant and cosine terms.
• Substitution: To handle more complex integrals like \( \int \cos(4x)dx \), substitution enabled a transformation of the variable to simplify the expression further.

These techniques are fundamental for solving integrals that initially seem too complex to tackle directly. They simplify problems and often lead to elegant solutions.

Substitution Method

The substitution method is a core technique in calculus used to simplify integrals by changing variables. This technique is akin to the reverse of the chain rule in differentiation. In our exercise, substitution was used to simplify the integration process:

• Introducing New Variables: We let \( u = 4x \) and derived that \( du = 4dx \). This transformation maps the intricate variable expressions into a simpler form.
• Adjusting Differential Elements: With \( dx = \frac{du}{4} \, \), the integral \( \int \cos(4x)dx \) becomes \( \frac{1}{32} \int \cos(u)du \).

By using substitution, we successfully transformed a cumbersome cosine term into a manageable integral. It simplifies integrations that involve complex functions or compositions, allowing them to be approached with more intuitive elementary rules of integration.