If f(x) is a function satisfying `f((1)/(x)) + x^(2) f(x) =0` for all non-zero x, then `int_(sin. theta)^(cosec theta) f(x)` dx equals

Options

A. `sin theta + cosec theta`
B. `sin^(2) theta`
C. `cosec^(2) theta`
D. 0

Answer

To solve the problem, we start with the given functional equation for \( f(x) \):

\[ f\left(\frac{1}{x}\right) + x^2 f(x) = 0 \]

### Step 1: Express \( f\left(\frac{1}{x}\right) \) in terms of \( f(x) \)

From the equation, we can isolate \( f\left(\frac{1}{x}\right) \):

\[ f\left(\frac{1}{x}\right) = -x^2 f(x) \]

### Step 2: Substitute \( x \) with \( \frac{1}{x} \)

Next, we substitute \( x \) with \( \frac{1}{x} \) in the original equation:

\[ f(x) + \left(\frac{1}{x}\right)^2 f\left(\frac{1}{x}\right) = 0 \]

This simplifies to:

\[ f(x) + \frac{1}{x^2} f\left(\frac{1}{x}\right) = 0 \]

### Step 3: Substitute \( f\left(\frac{1}{x}\right) \)

Now, we can substitute our expression for \( f\left(\frac{1}{x}\right) \) into this equation:

\[ f(x) + \frac{1}{x^2} \left(-x^2 f(x)\right) = 0 \]

This simplifies to:

\[ f(x) - f(x) = 0 \]

This is always true, which confirms that our functional equation is consistent.

### Step 4: Set up the integral

We need to evaluate the integral:

\[ I = \int_{\sin \theta}^{\csc \theta} f(x) \, dx \]

### Step 5: Change of variable

Using the substitution \( x = \frac{1}{t} \), we have \( dx = -\frac{1}{t^2} dt \). The limits change as follows:

- When \( x = \sin \theta \), \( t = \csc \theta \)
- When \( x = \csc \theta \), \( t = \sin \theta \)

Thus, we can rewrite the integral as:

\[ I = \int_{\csc \theta}^{\sin \theta} f\left(\frac{1}{t}\right) \left(-\frac{1}{t^2}\right) dt \]

This becomes:

\[ I = \int_{\sin \theta}^{\csc \theta} -\frac{1}{t^2} f\left(\frac{1}{t}\right) dt \]

### Step 6: Substitute \( f\left(\frac{1}{t}\right) \)

Using our earlier result \( f\left(\frac{1}{t}\right) = -t^2 f(t) \):

\[ I = \int_{\sin \theta}^{\csc \theta} -\frac{1}{t^2} \left(-t^2 f(t)\right) dt \]

This simplifies to:

\[ I = \int_{\sin \theta}^{\csc \theta} f(t) dt \]

### Step 7: Relate the two integrals

Now we have:

\[ I = \int_{\sin \theta}^{\csc \theta} f(x) \, dx \]

From our previous steps, we also have:

\[ I = -I \]

### Step 8: Solve for \( I \)

This implies:

\[ 2I = 0 \implies I = 0 \]

### Conclusion

Thus, the value of the integral is:

\[ \int_{\sin \theta}^{\csc \theta} f(x) \, dx = 0 \]