If the anti derivative of `int sin^4x/x dx` is `f(x)` then `int(sin^4(p+q)x)/x dx` in terms of `f(x)` is

Options

A. `f{(p+q)x}`
B. `(f{(p+q)x})/(p+q)`
C. `f{(p+q)x}(p+q)`
D. None of these

Answer

To solve the problem, we need to express the integral \(\int \frac{\sin^4((p+q)x)}{x} \, dx\) in terms of the function \(f(x)\), where \(f(x) = \int \frac{\sin^4 x}{x} \, dx\).

### Step-by-Step Solution:

1. **Understand the Given Information**:
We know that \(f(x) = \int \frac{\sin^4 x}{x} \, dx\).

2. **Change of Variables**:
We will perform a change of variables to express the integral in terms of \(f\). Let:
\[ t = (p + q)x \]
Then, differentiating both sides gives:
\[ dt = (p + q) \, dx \quad \Rightarrow \quad dx = \frac{dt}{p + q} \]

3. **Substituting in the Integral**:
Now we substitute \(x\) and \(dx\) in the integral:
\[ \int \frac{\sin^4((p+q)x)}{x} \, dx = \int \frac{\sin^4(t)}{\frac{t}{p + q}} \cdot \frac{dt}{p + q} \]
This simplifies to:
\[ = \int \frac{\sin^4(t)}{t} \, dt \]

4. **Relating to \(f(t)\)**:
From the definition of \(f(x)\), we know:
\[ f(t) = \int \frac{\sin^4(t)}{t} \, dt \]
Therefore, we can write:
\[ \int \frac{\sin^4(t)}{t} \, dt = f(t) \]

5. **Substituting Back for \(t\)**:
Now, we substitute back \(t = (p + q)x\):
\[ f(t) = f((p + q)x) \]

6. **Final Result**:
Thus, we have:
\[ \int \frac{\sin^4((p+q)x)}{x} \, dx = f((p + q)x) \]

### Conclusion:
The integral \(\int \frac{\sin^4((p+q)x)}{x} \, dx\) in terms of \(f(x)\) is:
\[ f((p + q)x) \]