Let a and b be two positive real numbers. Then the value of `int_(a)^(b)(e^(x//a)-e^(b//x))/(x)dx` is

Options

A. 0
B. ab
C. 1/ab
D. `e^(ab)`

Explanation

To solve the integral

\[ I = \int_{a}^{b} \frac{e^{\frac{x}{a}} - e^{\frac{b}{x}}}{x} \, dx, \]

we will use a substitution method.

### Step 1: Substitution
Let us substitute \( x = \frac{ab}{y} \). Then, we need to find \( dx \):

\[ dx = -\frac{ab}{y^2} \, dy. \]

### Step 2: Change the limits
When \( x = a \), \( y = \frac{ab}{a} = b \) and when \( x = b \), \( y = \frac{ab}{b} = a \). Thus, the limits of integration change from \( a \) to \( b \) to \( b \) to \( a \).

### Step 3: Substitute in the integral
Now, substituting \( x \) and \( dx \) into the integral:

\[ I = \int_{b}^{a} \frac{e^{\frac{ab/y}{a}} - e^{\frac{b}{ab/y}}}{\frac{ab}{y}} \left(-\frac{ab}{y^2}\right) \, dy. \]

This simplifies to:

\[ I = \int_{b}^{a} \left( e^{\frac{b}{y}} - e^{\frac{y}{a}} \right) \frac{y}{ab} \left(-\frac{ab}{y^2}\right) \, dy. \]

### Step 4: Simplify the integral
This can be simplified to:

\[ I = \int_{b}^{a} \left( e^{\frac{b}{y}} - e^{\frac{y}{a}} \right) \left(-\frac{1}{y}\right) \, dy. \]

### Step 5: Change the limits
Changing the limits of integration gives:

\[ I = -\int_{a}^{b} \left( e^{\frac{b}{y}} - e^{\frac{y}{a}} \right) \frac{1}{y} \, dy. \]

### Step 6: Combine the integrals
Now, we have:

\[ I = \int_{a}^{b} \frac{e^{\frac{x}{a}} - e^{\frac{b}{x}}}{x} \, dx = -I. \]

### Step 7: Solve for \( I \)
Adding \( I \) to both sides gives:

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

Thus, the value of the integral is

\[ \boxed{0}. \]