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In calculus, an antiderivative of a function is another function whose derivative equals the original function. Essentially, it's the reverse process of differentiation. When you've found an antiderivative, you have found one particular solution of a family of solutions that satisfy the original function when differentiated. With antiderivatives, there's usually a constant of integration, denoted as \( C \), because there are infinitely many antiderivatives for any given function which differ by a constant.

In our exercise, we're dealing with the function to integrate, \( 7 \sin \frac{\theta}{3} \). We aim to find the function where, if derived, results in this integral expression. By understanding each component of the function, especially constants and transformations, solving for the antiderivative becomes attainable.

Trigonometric integration involves finding the integral of functions involving trigonometric functions. This is common in calculus due to the pervasive nature of sinusoids in mathematical modeling, engineering, and physics.

In the given exercise, we need to integrate \( \sin \frac{\theta}{3} \). The basic rule for integrating sine functions is to know the antiderivative of \( \sin(u) \), which is \( -\cos(u) \). However, in this problem, the angle \( \theta \) is divided by \( 3 \), indicating that we need to consider this change of variable as part of the integral process – leading us to use substitution.

The substitution method is a technique used to simplify integrals, especially when they involve complicated functions. It is akin to the reverse chain rule in differentiation. The essence of substitution is to replace a part of the integral with a new variable to simplify the expression.

In the exercise, we choose \( u = \frac{\theta}{3} \) and thus \( du = \frac{1}{3} d\theta \), which yields \( d\theta = 3 \, du \). Therefore, the integral \( \int \sin \frac{\theta}{3} \, d\theta \) is transformed into \( 21 \int \sin(u) \, du \). This makes the problem more manageable, allowing us to apply known integral formulas effectively. Substitution requires careful attention to change the limits of integration if definite, or substitute back at the end when dealing with indefinite integrals.

Once you've found an antiderivative, it's crucial to verify your solution by differentiating it. This confirmation step ensures that the solution you derived leads back to the original function, reinforcing the correctness of your integration.

For our solution, \( -21 \cos\left( \frac{\theta}{3} \right) + C \), we differentiate with respect to \( \theta \). Doing so should yield the original integral, which is \( 7 \sin \frac{\theta}{3} \). Differentiation of \( \cos\left( \frac{\theta}{3} \right) \) leads to \( -\sin\left( \frac{\theta}{3} \right) \cdot \frac{1}{3} \) due to the chain rule. Therefore, multiplying by \( -21 \), we return to our initial function, thus confirming the correctness of our antiderivative.

A successful differentiation check serves as a solid confirmation that the integration process was executed correctly and that the substitution was properly reversed, reaffirming accuracy.