W(x)= ∫ 100 cos 4x dx = (k)sin (n)x +c. What are the values of k and n?

To solve the integral \( W(x) = \int 100 \cos 4x \, dx \) and express it in the form \( W(x) = k \sin(nx) + c \), we can follow these steps:

1. **Compute the integral:**
   \[
   W(x) = \int 100 \cos 4x \, dx
   \]
   To solve this, recognize that the integral of \( \cos(ax) \) is \( \frac{1}{a} \sin(ax) \).

   \[
   W(x) = 100 \cdot \int \cos 4x \, dx = 100 \cdot \left(\frac{1}{4} \sin 4x\right) + c
   \]

2. **Simplify the expression:**
   \[
   W(x) = 25 \sin 4x + c
   \]

Now, we can match this to the form \( k \sin(nx) + c \).

3. **Determine the values of \( k \) and \( n \):**

   From \( W(x) = 25 \sin 4x + c \), we can see that:
   - \( k = 25 \)
   - \( n = 4 \)

So, the values of \( k \) and \( n \) are \( \boxed{25} \) and \( \boxed{4} \), respectively.

Would you like more details on this process or have any questions?

Here are some related questions:
1. How do you integrate trigonometric functions with different coefficients?
2. What is the significance of the constant \( c \) in indefinite integrals?
3. How do you verify the result of an integral?
4. What happens if the integral involves \( \sin(ax) \) instead of \( \cos(ax) \)?
5. How do you find the derivative of the resulting function \( W(x) \)?

**Tip:** Always check your integration result by differentiating it to ensure you return to the original function.

Learn how to solve the integral ∫ 100 cos 4x dx = (k)sin (n)x +c with a detailed step-by-step solution. Understand the process of integrating trigonometric functions and expressing the result in a specific form. Suitable for advanced high school students and learners of integral calculus.

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