What is the integral of cos(3x) dx?

What is the integral of cos(3x) dx?

• A. (1/3)sin(3x) + C
• B. (1/2)sin(3x) + C
• C. (1/6)sin(3x) + C
• D. (3)sin(3x) + C

Answer: (A)

To find the integral of cos(3x) with respect to x, we apply the rule for integrating cosine functions. The integral of cos(kx) is given by (1/k)sin(kx) + C, where k is a constant and C represents the constant of integration. In this case, k is equal to 3. Therefore, we can substitute this value into the formula: \[ \int \cos(3x) \, dx = \frac{1}{3} \sin(3x) + C. \] This shows that the correct answer involves taking the derivative of the inner function (3x) which contributes a factor of 3, and thus we must divide by that factor when integrating. Thus, the integral is \(\frac{1}{3} \sin(3x) + C\), confirming that the provided response is accurate. This result reflects not only a correct application of the integration rule for cosines but also adherence to the necessary adjustments associated with the linear transformation of the argument of the cosine function.

To find the integral of cos(3x) with respect to x, we apply the rule for integrating cosine functions. The integral of cos(kx) is given by (1/k)sin(kx) + C, where k is a constant and C represents the constant of integration.

In this case, k is equal to 3. Therefore, we can substitute this value into the formula:

[

\int \cos(3x) , dx = \frac{1}{3} \sin(3x) + C.

]

This shows that the correct answer involves taking the derivative of the inner function (3x) which contributes a factor of 3, and thus we must divide by that factor when integrating.

Thus, the integral is (\frac{1}{3} \sin(3x) + C), confirming that the provided response is accurate. This result reflects not only a correct application of the integration rule for cosines but also adherence to the necessary adjustments associated with the linear transformation of the argument of the cosine function.