To evaluate the integral \(\int \sin 3x \, \mathrm{d}x\), we will use the technique of substitution. Let's go through the steps in detail. ### Step 1: Identify the Integral We need to evaluate: \[ \int \sin 3x \, \mathrm{d}x \] ### Step 2: Use Substitution To simplify the integration, we can use a substitution. Let: \[ u = 3x \] Then, we differentiate \(u\) with respect to \(x\): \[ \frac{\mathrm{d}u}{\mathrm{d}x} = 3 \quad \Rightarrow \quad \mathrm{d}u = 3 \, \mathrm{d}x \quad \Rightarrow \quad \mathrm{d}x = \frac{1}{3} \mathrm{d}u \] ### Step 3: Substitute in the Integral Now, we can substitute \(u\) and \(\mathrm{d}x\) into the integral: \[ \int \sin 3x \, \mathrm{d}x = \int \sin u \cdot \frac{1}{3} \mathrm{d}u = \frac{1}{3} \int \sin u \, \mathrm{d}u \] ### Step 4: Integrate \(\sin u\) The integral of \(\sin u\) is: \[ \int \sin u \, \mathrm{d}u = -\cos u + C \] where \(C\) is the constant of integration. ### Step 5: Substitute Back Now, we substitute back \(u = 3x\): \[ \frac{1}{3} \int \sin u \, \mathrm{d}u = \frac{1}{3} (-\cos u + C) = -\frac{1}{3} \cos(3x) + C \] ### Final Answer Thus, the evaluated integral is: \[ \int \sin 3x \, \mathrm{d}x = -\frac{1}{3} \cos(3x) + C \] ### Conclusion The correct option is: **C. (-1/3) cos 3x + c** ### Explanation of Other Options - **Option A: (2/3) cos 3x + c** - This option is incorrect because it suggests a positive coefficient for \(\cos(3x)\), which contradicts the integral of \(\sin(3x)\) that results in a negative cosine function. - **Option B: (1/3) cos 3x + c** - This option is also incorrect for the same reason as option A. The integral of \(\sin(3x)\) should yield a negative cosine term, not a positive one. - **Option D: (-2/3) cos 3x + c** - This option is incorrect because it has the wrong coefficient. The correct coefficient is \(-\frac{1}{3}\), not \(-\frac{2}{3}\). ### Revision Summary - Use substitution to simplify integrals involving trigonometric functions. - Remember that the integral of \(\sin u\) is \(-\cos u + C\). - Always substitute back to the original variable after integration. - Check the signs and coefficients carefully when selecting the correct answer from multiple choices.