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Trigonometric integrals are a special type of integral where the integrand includes trigonometric functions like sine or cosine. These integrals often require specific techniques for solving.
In our example, the function is \( \int \sin(3x) \, dx \) which falls under this category. We focus on integrals where a variable is inside the sine function, with a constant multiplied by it. Understanding how to work with trigonometric identities can significantly simplify the process of finding the integral.
Remember, trigonometric integrals often use formulas derived from derivative rules. Knowing these can help you solve them quickly and accurately.

An antiderivative is essentially the reverse process of differentiation. Finding an antiderivative means determining a function whose derivative is the original function you started with. This concept is crucial when working with integrals. For the problem \( \int \sin(3x) \, dx \), we look for an antiderivative of \( \sin(3x) \). Using the standard formula for sine, we find: \[ \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \] Applying this, our antiderivative becomes: \(-\frac{1}{3} \cos(3x) + C\). This result provides us the expression needed to evaluate the definite integral by assigning integration limits.

Integration limits define the range over which a definite integral is evaluated. These limits transform an indefinite integral, which includes the constant of integration, into a specific numerical result.For our integral, the limits are from 0 to \( \frac{\pi}{3} \). We substitute these limits into the antiderivative—calculated previously—to find the precise area under the curve of the function \( \sin(3x) \) within the specified domain. Proper use of limits allows converting a generalized antiderivative into a tangible result. Always substitute these limits carefully according to their position, upper or lower, in the integral expression.

Evaluating the definite integral \( \int_{0}^{\pi / 3} \sin(3x) \, dx \) requires applying the antiderivative formula alongside the set integration limits.
First, substitute the upper limit into the antiderivative: \[ \left[-\frac{1}{3} \cos\left(3 \times \frac{\pi}{3}\right)\right] \] This equals \(-\frac{1}{3} \cos(\pi)\), giving \(\frac{1}{3}\). Next, do the same for the lower limit: \(-\frac{1}{3} \cos(3 \times 0)\), simplifying to \(-\frac{1}{3}\). Finally, subtract these two results, following the formula \(\int_a^b f(x)dx = F(b) - F(a)\), to obtain: \(\frac{1}{3} - (-\frac{1}{3}) = \frac{2}{3}\). This process concludes the evaluation by providing the exact area under the curve as determined by the definite integral.