A function f(x) satisfie `f(x)=f((c)/(x))` for some real number c( gt 1) and all positive number 'x'. If `int_(1)^(sqrtc)(f(x))/(x)dx=3`, then `int_(1)^(c)(f(x))/(x)dx` is

Options

A. 4
B. 6
C. 8
D. 9

Explanation

To solve the problem step by step, we will use the properties of the function \( f(x) \) and the given integral.

### Step 1: Understanding the Function
We are given that \( f(x) = f\left(\frac{c}{x}\right) \) for some real number \( c > 1 \) and for all positive \( x \). This property implies that the function \( f(x) \) is symmetric with respect to the transformation \( x \rightarrow \frac{c}{x} \).

### Step 2: Given Integral
We know that:
\[ \int_{1}^{\sqrt{c}} \frac{f(x)}{x} \, dx = 3 \]

### Step 3: Splitting the Integral
We want to find:
\[ \int_{1}^{c} \frac{f(x)}{x} \, dx \]
We can split this integral into two parts:
\[ \int_{1}^{c} \frac{f(x)}{x} \, dx = \int_{1}^{\sqrt{c}} \frac{f(x)}{x} \, dx + \int_{\sqrt{c}}^{c} \frac{f(x)}{x} \, dx \]

### Step 4: Changing Variables in the Second Integral
For the second integral, we will use the substitution \( x = \frac{c}{t} \). Then, we have:
\[ dx = -\frac{c}{t^2} \, dt \]
When \( x = \sqrt{c} \), \( t = \sqrt{c} \) and when \( x = c \), \( t = 1 \). Thus, the limits change from \( \sqrt{c} \) to \( 1 \).

Now, substituting into the integral:
\[ \int_{\sqrt{c}}^{c} \frac{f(x)}{x} \, dx = \int_{\sqrt{c}}^{1} \frac{f\left(\frac{c}{t}\right)}{\frac{c}{t}} \left(-\frac{c}{t^2}\right) \, dt \]
This simplifies to:
\[ \int_{1}^{\sqrt{c}} \frac{f\left(\frac{c}{t}\right)}{t} \, dt \]
Using the property of the function \( f(x) \):
\[ f\left(\frac{c}{t}\right) = f(t) \]
Thus,
\[ \int_{\sqrt{c}}^{c} \frac{f(x)}{x} \, dx = \int_{1}^{\sqrt{c}} \frac{f(t)}{t} \, dt \]

### Step 5: Combining the Integrals
Now, we can combine the two integrals:
\[ \int_{1}^{c} \frac{f(x)}{x} \, dx = \int_{1}^{\sqrt{c}} \frac{f(x)}{x} \, dx + \int_{1}^{\sqrt{c}} \frac{f(t)}{t} \, dt \]
This gives us:
\[ \int_{1}^{c} \frac{f(x)}{x} \, dx = 3 + 3 = 6 \]

### Final Answer
Thus, we find that:
\[ \int_{1}^{c} \frac{f(x)}{x} \, dx = 6 \]