Problem 56 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate \(\int \sin ^{2} x d x\).

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{x}{2} - \frac{\sin(2x)}{4} + C \).

Step 1: Use Trigonometric Identity

To evaluate the integral of \( \sin^2 x \), we first use the trigonometric identity: \( \sin^2 x = \frac{1 - \cos(2x)}{2} \). This transformation helps us simplify the process of integration by avoiding the need to integrate the square directly.

Step 2: Substitute into the Integral

With the identity, we substitute \( \sin^2 x \) in the integral: \[ \int \sin^2 x \, dx = \int \frac{1 - \cos(2x)}{2} \, dx. \]This allows us to split the integral into simpler parts: \[ \frac{1}{2} \int (1 - \cos(2x)) \, dx. \]

Step 3: Integrate Each Term

Now, integrate each term separately. For \( \int 1 \, dx \), it's simply \( x \). For \( \int \cos(2x) \, dx \), use the substitution rule. Let \( u = 2x \), then \( du = 2 \, dx \) or \( dx = \frac{1}{2} du \), so\[ \int \cos(2x) \, dx = \frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) = \frac{1}{2} \sin(2x). \]

Step 4: Combine the Results

Combine the results from Step 3 to find:\[ \frac{1}{2}(x) - \frac{1}{2} \left( \frac{1}{2} \sin(2x) \right) + C = \frac{x}{2} - \frac{\sin(2x)}{4} + C, \]where \( C \) is the constant of integration.

Key Concepts

Trigonometric Integration

Trigonometric integration deals with the process of integrating functions that involve trigonometric expressions. This can often seem daunting, especially with more complex functions like \( \sin^2 x \) or \( \cos^2 x \). However, these integrals can be simplified using specific techniques. By applying a trigonometric identity, we transform a seemingly complicated integral into a form that is easier to work with. For instance, the integral \( \int \sin^2 x \, dx \) can be simplified by using the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \). This substitution simplifies the expression and makes the integration process much more straightforward, allowing us to focus on more elementary functions. Trigonometric integration is a powerful tool, especially useful in handling problems that involve oscillations and waves.

Integration Techniques

Integration techniques are essential tools used to solve integrals that are not straightforward. One common technique demonstrated in this integral is the use of trigonometric identities to simplify the integrand. Once simplified, the process often involves splitting the integral into parts that are easier to handle separately.
In the example of \( \int \sin^2 x \, dx \), after applying the identity, the integral becomes \( \int \frac{1 - \cos(2x)}{2} \, dx \). This can be rewritten as \( \frac{1}{2} \int 1 \, dx - \frac{1}{2} \int \cos(2x) \, dx \).
Each of these integrals can now be tackled individually:

• The first part, \( \frac{1}{2} \int 1 \, dx \), simplifies to \( \frac{x}{2} \).
• The second part requires a substitution, such as \( u = 2x \), which turns \( \int \cos(2x) \, dx \) into a simpler form, \( \frac{1}{2} \sin(2x) \).

Combining these results gives the final answer easily, showcasing the usefulness of integration techniques.

Trigonometric Identities

Trigonometric identities are equations that relate various trigonometric functions to one another. They are invaluable in simplifying complex trigonometric expressions, making them easier to work with, especially in calculus.
In solving the integral \( \int \sin^2 x \, dx \), the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) played a crucial role. This identity allowed us to transform \( \sin^2 x \) into an expression involving \( \cos(2x) \), a function that is simpler to integrate. By breaking down trigonometric expressions using identities, we can streamline the process of integration significantly.
These identities are not just limited to \( \sin^2 x \); numerous other identities exist, such as double angle identities, sum-to-product identities, and more, all of which can be creatively applied to simplify various mathematical problems. Understanding and using these identities is essential for success in integral calculus, transforming complex integrals into digestible pieces.