`int _(log pi - log 2 ) ^(log pi) (e ^(x))/(1- cos ((2)/(3)e ^(x)))dx ` is equal to

Options

A. `sqrt3`
B. `-sqrt3`
C. `(1)/( sqrt3)`
D. `-(1)/(sqrt3)`

Answer

To solve the integral

\[ I = \int_{\log \pi}^{\log 2} \frac{e^x}{1 - \cos\left(\frac{2}{3} e^x\right)} \, dx, \]

we can follow these steps:

### Step 1: Substitution
Let

\[ t = \frac{2}{3} e^x. \]

Then, differentiating both sides gives us:

\[ \frac{dt}{dx} = \frac{2}{3} e^x \implies dt = \frac{2}{3} e^x \, dx \implies e^x \, dx = \frac{3}{2} dt. \]

### Step 2: Change of Limits
Now, we need to change the limits of integration.

- When \( x = \log \pi \):

\[ t = \frac{2}{3} e^{\log \pi} = \frac{2\pi}{3}. \]

- When \( x = \log 2 \):

\[ t = \frac{2}{3} e^{\log 2} = \frac{2 \cdot 2}{3} = \frac{4}{3}. \]

Thus, the limits change from \( x: [\log \pi, \log 2] \) to \( t: \left[\frac{2\pi}{3}, \frac{4}{3}\right] \).

### Step 3: Rewrite the Integral
Now we can rewrite the integral \( I \):

\[ I = \int_{\frac{2\pi}{3}}^{\frac{4}{3}} \frac{\frac{3}{2} dt}{1 - \cos t}. \]

### Step 4: Factor Out Constants
We can factor out the constant \( \frac{3}{2} \):

\[ I = \frac{3}{2} \int_{\frac{2\pi}{3}}^{\frac{4}{3}} \frac{dt}{1 - \cos t}. \]

### Step 5: Simplifying the Denominator
Using the identity \( 1 - \cos t = 2 \sin^2\left(\frac{t}{2}\right) \):

\[ I = \frac{3}{2} \int_{\frac{2\pi}{3}}^{\frac{4}{3}} \frac{dt}{2 \sin^2\left(\frac{t}{2}\right)} = \frac{3}{4} \int_{\frac{2\pi}{3}}^{\frac{4}{3}} \csc^2\left(\frac{t}{2}\right) dt. \]

### Step 6: Integration of Cosecant Squared
The integral of \( \csc^2 u \) is \( -\cot u \):

\[ I = \frac{3}{4} \left[-\cot\left(\frac{t}{2}\right)\right]_{\frac{2\pi}{3}}^{\frac{4}{3}}. \]

### Step 7: Evaluate the Limits
Now we evaluate the limits:

\[ I = \frac{3}{4} \left[-\cot\left(\frac{4}{6}\right) + \cot\left(\frac{2\pi}{3}\right)\right]. \]

Calculating \( \cot\left(\frac{2\pi}{3}\right) = -\frac{1}{\sqrt{3}} \) and \( \cot\left(\frac{4}{6}\right) = \cot\left(\frac{2\pi}{3}\right) = -\frac{1}{\sqrt{3}} \).

Thus,

\[ I = \frac{3}{4} \left[-(-\frac{1}{\sqrt{3}}) + \left(-\frac{1}{\sqrt{3}}\right)\right] = \frac{3}{4} \left[\frac{1}{\sqrt{3}} - \frac{1}{\sqrt{3}}\right] = 0. \]

### Final Answer
Thus, the value of the integral is

\[ \boxed{0}. \]