If `intcos2018xsin^(2016)x dx=f(x)+c`, where `f(0)=0`, then

Options

A. `int_(0)^(pi)f(x)dx=f(pi)`
B. `f((pi/4)=1/(2^(1009)) . 1/2017`
C. `f((pi)/4)=1/((2018)2^(1009))`
D. `f(x)` is odd function

Explanation

To solve the problem, we need to evaluate the integral \( \int \cos(2018x) \sin^{2016}(x) \, dx \) and express it in the form \( f(x) + c \) where \( f(0) = 0 \).

### Step-by-Step Solution:

1. **Identify the Integral**:
We start with the integral:
\[ I = \int \cos(2018x) \sin^{2016}(x) \, dx \]

2. **Use Integration by Parts**:
We can use integration by parts. Let:
- \( u = \sin^{2016}(x) \)
- \( dv = \cos(2018x) \, dx \)

Then, we need to find \( du \) and \( v \):
- \( du = 2016 \sin^{2015}(x) \cos(x) \, dx \)
- \( v = \frac{1}{2018} \sin(2018x) \)

3. **Apply Integration by Parts Formula**:
The integration by parts formula is:
\[ \int u \, dv = uv - \int v \, du \]
Applying this, we get:
\[ I = \left( \sin^{2016}(x) \cdot \frac{1}{2018} \sin(2018x) \right) - \int \frac{1}{2018} \sin(2018x) \cdot 2016 \sin^{2015}(x) \cos(x) \, dx \]

4. **Simplify the Integral**:
This simplifies to:
\[ I = \frac{1}{2018} \sin^{2016}(x) \sin(2018x) - \frac{2016}{2018} \int \sin^{2015}(x) \cos(x) \sin(2018x) \, dx \]

5. **Evaluate the Remaining Integral**:
The remaining integral can be evaluated using similar techniques or further integration by parts, but for our purpose, we will denote it as \( J \):
\[ J = \int \sin^{2015}(x) \cos(x) \sin(2018x) \, dx \]

6. **Final Expression**:
Thus, we can express the original integral as:
\[ I = \frac{1}{2018} \sin^{2016}(x) \sin(2018x) - \frac{2016}{2018} J + C \]
where \( C \) is the constant of integration.

7. **Determine \( f(x) \)**:
To find \( f(x) \), we need to ensure that \( f(0) = 0 \). Evaluating at \( x = 0 \):
\[ f(0) = \frac{1}{2018} \sin^{2016}(0) \sin(0) - \frac{2016}{2018} J(0) + C = 0 \]
Since both \( \sin^{2016}(0) \) and \( \sin(0) \) are 0, we have:
\[ C = 0 \]
Therefore, \( f(x) \) can be expressed as:
\[ f(x) = \frac{1}{2018} \sin^{2016}(x) \sin(2018x) - \frac{2016}{2018} J \]