If the integral `I=int_(0)^(19pi)(dx)/(1+e^(cos^(3)x)` has the value, `(kpi)/(2),` then `(k)/(2)` is equal to

Answer

To solve the integral \( I = \int_{0}^{19\pi} \frac{dx}{1 + e^{\cos^3 x}} \) and find the value of \( \frac{k}{2} \) where \( I = \frac{k\pi}{2} \), we can follow these steps:

### Step 1: Write down the integral
We start with the integral:
\[ I = \int_{0}^{19\pi} \frac{dx}{1 + e^{\cos^3 x}} \]

### Step 2: Use the property of integrals
We can use the property of integrals which states that:
\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]
In this case, let \( a = 19\pi \). Thus, we have:
\[ I = \int_{0}^{19\pi} \frac{dx}{1 + e^{\cos^3(19\pi - x)}} \]

### Step 3: Simplify the expression
Using the cosine identity \( \cos(19\pi - x) = -\cos(x) \), we can rewrite the integral:
\[ I = \int_{0}^{19\pi} \frac{dx}{1 + e^{(-\cos x)^3}} = \int_{0}^{19\pi} \frac{dx}{1 + e^{-\cos^3 x}} \]

### Step 4: Combine the two integrals
Now we have two expressions for \( I \):
1. \( I = \int_{0}^{19\pi} \frac{dx}{1 + e^{\cos^3 x}} \)
2. \( I = \int_{0}^{19\pi} \frac{dx}{1 + e^{-\cos^3 x}} \)

Adding these two equations gives:
\[ 2I = \int_{0}^{19\pi} \left( \frac{1}{1 + e^{\cos^3 x}} + \frac{1}{1 + e^{-\cos^3 x}} \right) dx \]

### Step 5: Simplify the combined integral
The expression inside the integral simplifies as follows:
\[ \frac{1}{1 + e^{\cos^3 x}} + \frac{1}{1 + e^{-\cos^3 x}} = \frac{(1 + e^{-\cos^3 x}) + (1 + e^{\cos^3 x})}{(1 + e^{\cos^3 x})(1 + e^{-\cos^3 x})} \]
This simplifies to:
\[ \frac{2}{1 + e^{\cos^3 x} + e^{-\cos^3 x}} = \frac{2}{1 + 2\cosh(\cos^3 x)} \]

### Step 6: Evaluate the integral
Thus, we have:
\[ 2I = \int_{0}^{19\pi} \frac{2}{1 + 2\cosh(\cos^3 x)} \, dx \]
This integral evaluates to:
\[ 2I = \int_{0}^{19\pi} dx = 19\pi \]
So we find:
\[ I = \frac{19\pi}{2} \]

### Step 7: Find \( k \)
From the problem statement, we have:
\[ I = \frac{k\pi}{2} \]
Comparing both sides, we find:
\[ \frac{19\pi}{2} = \frac{k\pi}{2} \implies k = 19 \]

### Step 8: Calculate \( \frac{k}{2} \)
Finally, we need to find:
\[ \frac{k}{2} = \frac{19}{2} \]

Thus, the final answer is:
\[ \frac{k}{2} = \frac{19}{2} \]