If `int_(0)^(x)f(x)sint dt=" constant, " 0 lt x lt 2pi and f(pi)=2`, then the value of `f(pi//2)` is

Options

A. 3
B. 2
C. 4
D. 8

Explanation

To solve the problem, we need to find the value of \( f\left(\frac{\pi}{2}\right) \) given that

\[ \int_{0}^{x} f(x) \sin t \, dt = \text{constant} \]

for \( 0 < x < 2\pi \) and \( f(\pi) = 2 \).

### Step 1: Differentiate both sides with respect to \( x \)

We start by differentiating both sides of the equation with respect to \( x \):

\[ \frac{d}{dx} \left( \int_{0}^{x} f(x) \sin t \, dt \right) = 0 \]

Using the Leibniz rule for differentiation under the integral sign, we have:

\[ f(x) \sin x + \int_{0}^{x} f'(x) \sin t \, dt = 0 \]

### Step 2: Simplify the equation

From the above equation, we can isolate \( f'(x) \):

\[ f(x) \sin x = -\int_{0}^{x} f'(x) \sin t \, dt \]

### Step 3: Rearranging the terms

Rearranging gives us:

\[ f'(x) \int_{0}^{x} \sin t \, dt = -f(x) \sin x \]

### Step 4: Evaluate the integral

The integral \( \int_{0}^{x} \sin t \, dt \) evaluates to:

\[ -\cos x + 1 \]

Thus, we can rewrite the equation as:

\[ f'(x)(1 - \cos x) = -f(x) \sin x \]

### Step 5: Separate variables

Now we can separate variables:

\[ \frac{f'(x)}{f(x)} = -\frac{\sin x}{1 - \cos x} \]

### Step 6: Integrate both sides

Integrating both sides gives:

\[ \ln |f(x)| = -\int \frac{\sin x}{1 - \cos x} \, dx \]

To integrate the right side, we can use the substitution \( u = 1 - \cos x \), \( du = \sin x \, dx \):

\[ \int \frac{\sin x}{1 - \cos x} \, dx = -\ln |1 - \cos x| + C \]

Thus, we have:

\[ \ln |f(x)| = \ln |C| - \ln |1 - \cos x| \]

This simplifies to:

\[ f(x) = \frac{C}{1 - \cos x} \]

### Step 7: Use the given condition \( f(\pi) = 2 \)

Substituting \( x = \pi \):

\[ f(\pi) = \frac{C}{1 - \cos \pi} = \frac{C}{1 - (-1)} = \frac{C}{2} \]

Setting this equal to 2 gives:

\[ \frac{C}{2} = 2 \implies C = 4 \]

### Step 8: Write the function \( f(x) \)

Thus, we have:

\[ f(x) = \frac{4}{1 - \cos x} \]

### Step 9: Find \( f\left(\frac{\pi}{2}\right) \)

Now we can find \( f\left(\frac{\pi}{2}\right) \):

\[ f\left(\frac{\pi}{2}\right) = \frac{4}{1 - \cos\left(\frac{\pi}{2}\right)} = \frac{4}{1 - 0} = 4 \]

### Final Answer

The value of \( f\left(\frac{\pi}{2}\right) \) is:

\[ \boxed{4} \]