Question

Perform the indicated integrations. $$ \int \sin ^{2} x d x $$

Short answer

The integral of \( \sin^2 x \) is \( \frac{1}{2} x - \frac{1}{4} \sin(2x) + C \).

Step-by-step solution

Use Trigonometric Identity

To integrate \( \sin^2 x \), we can use the trigonometric identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \). This identity will help simplify the integration.

Rewrite the Integral

Using the trigonometric identity, rewrite the integral as follows: \[ \int \sin^2 x \, dx = \int \frac{1 - \cos(2x)}{2} \, dx \] This simplifies the problem to integrating \( \frac{1}{2} \int 1 \, dx - \frac{1}{2} \int \cos(2x) \, dx \).

Integrate the Constant Term

The integral of the constant \(1\) is straightforward: \[ \frac{1}{2} \int 1 \, dx = \frac{1}{2} x \] This is one part of our solution.

Integrate the Trigonometric Function

Now, integrate \( \cos(2x) \). Using the substitution method, recall that the integral of \( \cos(ax) \) is \( \frac{1}{a} \sin(ax) + C \). Thus: \[ \frac{1}{2} \int \cos(2x) \, dx = \frac{1}{2} \cdot \frac{1}{2} \sin(2x) = \frac{1}{4} \sin(2x) \] Combine this with the other part of the integral.

Combine and Write the Final Solution

Combine the results from Step 3 and Step 4: \[ \int \sin^2 x \, dx = \frac{1}{2} x - \frac{1}{4} \sin(2x) + C \] Where \( C \) is the integration constant.

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for different values of the involved variables. They are especially helpful in simplifying integrals, as shown in our exercise. In this case, the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) is used.
This particular identity helps transform the integral of \( \sin^2 x \) into a more manageable form, involving simpler integrable terms. Understanding these identities allows solving complex trigonometric integrals by turning them into easier expressions.
Here are a few key trigonometric identities that could be useful in integration and other calculus problems:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Double Angle Formulas: \( \sin(2x) = 2\sin(x)\cos(x) \) and \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

Utilizing these identities correctly is a foundational skill for calculus problem solving.

Substitution Method

The substitution method is a technique for evaluating integrals and simplifying complex expressions into more solvable forms. In the exercise, it is used to integrate \( \cos(2x) \). The substitution technique involves changing the variable of integration to make the integral more straightforward.
For the integral \( \frac{1}{2} \int \cos(2x) \, dx \), we use the substitution method to let \( u = 2x \), making \( du = 2 \, dx \) or \( dx = \frac{1}{2} \, du \). This transforms the integral into a simpler form:

• \( \frac{1}{2} \int \cos(u) \, \frac{1}{2} \, du \)

The key steps in the substitution method involve:

• Selecting a substitution that can simplify the integral.
• Changing the limits of integration if the integral is definite.
• Substituting back to the original variable after integration is complete.

This method is powerful for solving integrals involving composite functions, transforming them into basic integrals.

Definite and Indefinite Integrals

Integrals in calculus can be classified as definite or indefinite, each serving distinct purposes in analysis. In our work, we focused on an indefinite integral.
An indefinite integral, represented as \( \int f(x) \, dx \), yields the antiderivative and a family of functions, expressed with an integration constant \( C \). This represents the set of all possible antiderivatives of the function.
On the other hand, definite integrals involve computing the area under a curve from one point to another. It is expressed as \( \int_{a}^{b} f(x) \, dx \), resulting in a number, not a function.
The exercise demonstrated indefinite integration by calculating \( \int \sin^2 x \, dx \), leading to \( \frac{1}{2} x - \frac{1}{4} \sin(2x) + C \). This solution includes:

• The antiderivative of each term in the integral.
• The integration constant \( C \), indicating the family of antiderivatives.

Understanding the difference between these integrals enriches the problem-solving toolkit in calculus.

Calculus Problem Solving

Calculus problem solving involves various strategies and methods for tackling mathematical problems related to change and accumulation. The process typically includes setting up the problem, using appropriate techniques, and interpreting the results.
In the presented exercise, solving \( \int \sin^2 x \, dx \) required the combination of trigonometric identities, substitution, and a clear interpretation of indefinite integrals.
Key strategies for effective calculus problem solving include:

• Recognizing patterns and selecting tools such as identities and substitution methods.
• Breaking down complex problems into simpler parts.
• Verifying solutions and understanding their meaning in context.

Each of these steps plays a critical role in successfully navigating calculus problems, ensuring that solutions are both accurate and applicable in various contexts. Building these skills enhances confidence and capability in tackling advanced mathematical challenges.