Question

Verify each of the following answers for an indefinite integral by one or more of the methods suggested above. \(\int 2 \sin \theta \cos \theta d \theta=\sin ^{2} \theta \quad\) or \(\quad-\cos ^{2} \theta \quad\) or \(\quad-\frac{1}{2} \cos 2 \theta .\) Hint: Use trig identities.

Short answer

The correct solution is \(-\frac{1}{2} \cos 2 \theta\).

Step-by-step solution

Identify the integral form and proposed solutions

The given integral is \(\int 2 \sin \theta \cos \theta \ d\theta\). The proposed solutions are \(\sin^2 \theta\), \(-\cos^2 \theta\), and \(-\frac{1}{2} \cos 2 \theta\).

Simplify the integral using Trigonometric Identity

Use the double-angle identity \( \sin 2 \theta = 2 \sin \theta \cos \theta \) to rewrite the integral as \(\int \sin 2 \theta \ d\theta\).

Integrate

The integral of \(\sin 2 \theta\) is \(\-\frac{1}{2} \cos 2 \theta\). Thus, \(\int \sin 2 \theta \ d\theta = \int 2 \sin \theta \cos \theta \ d\theta = \-\frac{1}{2} \cos 2 \theta\).

Verify the solutions

Among the proposed solutions, \(-\frac{1}{2} \cos 2 \theta\) matches our integral calculation. \(\sin^2 \theta\) and \(-\cos^2 \theta\) do not match the result obtained from integration.

Key concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They are crucial in simplifying and solving integrals involving trigonometric functions.

• In this exercise, we employ the double-angle identity: \( \sin 2 \theta = 2 \sin \theta \cos \theta \)
• This identity helps to simplify the integral \(\int 2 \sin \theta \cos \theta d \theta\).
  
  Knowing and applying trigonometric identities can transform complex integrals into more familiar forms, making them easier to solve. This skill is essential in integral calculus, where the integrand often requires simplification before integration.

Integral Calculus

Integral calculus deals with the concept of \( \int integrals \), which represent areas under curves.

It includes indefinite integrals, which are essentially the antiderivatives of functions. In this context, the integral \(\int 2 \sin \theta \cos \theta d \theta\) is used to find a function whose derivative is \(2 \sin \theta \cos \theta \).

Here are some key steps:

• Identify the given integral and proposed solutions.
• Simplify using appropriate identities.
• Perform the integration.
• Verify the answers by differentiating them to see if you retrieve the original integrand.
  
  Understanding these steps helps in solving and verifying indefinite integrals. Getting comfortable with these concepts is fundamental to solving more complicated integral problems.

Double-Angle Formula

The double-angle formula is a specific trigonometric identity used for simplifying the product of sine and cosine functions. In this exercise, we use the formula: \(\sin 2 \theta = 2 \sin \theta \cos \theta\), to transform the integral into a more manageable form.

This formula is not just a simplification tool but also provides a bridge to solve integrals involving trigonometric functions. Here’s how it applies in steps:

• Identify \(2 \sin \theta \cos \theta \) in the integral.
• Use the identity \(\sin 2 \theta = 2 \sin \theta \cos \theta\) to rewrite the integral as \(\int \sin 2 \theta d \theta \).
• Integrate \(\sin 2 \theta \) to find the indefinite integral.
  
  This method simplifies the original integral and makes solving the indefinite integral straightforward. It’s a powerful tool in integral calculus, particularly for trigonometric integrals.

Mathematical Verification

Mathematical verification ensures the accuracy of a solution. In integral calculus, this usually involves differentiating the result to check if it matches the original integrand.

To verify our solution to \(\int 2 \sin \theta \cos \theta d \theta\): \(\-\frac{1}{2} \cos 2 \theta \)::

• Differentiate \(\-\frac{1}{2} \cos 2 \theta \)
• If the derivative matches the original function inside the integral, \( 2 \sin \theta \cos \theta \), the answer is correct.
  
  Verification confirms that our integral calculation aligns with the function's antiderivative. It’s a crucial step in ensuring solution accuracy in mathematics.