91Ó°ÊÓ

Trigonometric identities are essential tools in mathematics, especially when dealing with trigonometric integrals. They help in transforming complex expressions into simpler forms, making integration doable. In our example, we utilize the identity:\[ \cot 3x = \frac{\cos 3x}{\sin 3x} \].
This identity allows us to express the function \( \sin 3x \cot 3x \) in terms of simpler trigonometric functions. This step is crucial because integrating simpler functions is typically more straightforward. By substituting the identity, the integral becomes \( \int \cos 3x \, dx \), significantly reducing its complexity. Knowing and correctly applying trigonometric identities can make a huge difference in solving trigonometric integrals efficiently.

Understanding basic integration techniques is vital for evaluating integrals like the one in this exercise. After we apply the trigonometric identity, the problem reduces to integrating \( \cos 3x \). Recognizing standard forms is key here.

• The integral of \( \cos kx \) is \( \frac{1}{k} \sin kx + C \).
• This form corresponds to simple adjustments of known integral formulas for trigonometric functions.

Once the expression is simplified, identifying the scaling factor (in this case, 3) is necessary before integrating. Utilize the antiderivative directly to find \( \int \cos 3x \, dx = \frac{1}{3} \sin 3x + C \). This is a reflection of the technique called 'integration by recognition,' useful for common trigonometric integrals.

In calculus, when integrating a function, the result always includes an arbitrary constant of integration, denoted as \( C \). This constant represents the family of all antiderivatives, due to the fact that derivative of a constant is zero.
So even though derivatives can pinpoint the slope of a tangent at any specific point, indefinite integrals lead to general forms with \( C \) to cover all potential vertical shifts of the function. This principle applies in our result \( \frac{1}{3} \sin 3x + C \), where the addition of \( C \) acknowledges an infinite number of valid antiderivatives. Always remember to include this constant to reflect all possible solutions accurately.