If the minimum of the following function `f(x)` defined at ` 0 lt x lt pi/2`.
`f(x) = int_(0)^(x)(d theta)/(cos theta) + int_(x)^(pi//2) (d theta)/(sin theta)` is equal to `ln(a+sqrt(b))` where a, b `in` N and b is not perfet square then find value of `(a+b)`

Answer

To solve the problem, we start by analyzing the function defined by the integrals:

\[ f(x) = \int_{0}^{x} \frac{d\theta}{\cos \theta} + \int_{x}^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta} \]

### Step 1: Evaluate the first integral
The first integral can be evaluated as follows:

\[ \int_{0}^{x} \frac{d\theta}{\cos \theta} = \ln(\sec x + \tan x) \quad \text{(using the integral of } \frac{1}{\cos \theta} \text{)} \]

### Step 2: Evaluate the second integral
The second integral can be evaluated similarly:

\[ \int_{x}^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta} = -\ln(\cos \theta) \bigg|_{x}^{\frac{\pi}{2}} = -\ln(0) + \ln(\cos x) = \ln(\cos x) \quad \text{(since } \cos(\frac{\pi}{2}) = 0\text{)} \]

### Step 3: Combine the results
Now we combine the results of both integrals:

\[ f(x) = \ln(\sec x + \tan x) + \ln(\cos x) \]

Using the property of logarithms, we can combine these:

\[ f(x) = \ln((\sec x + \tan x) \cdot \cos x) \]

### Step 4: Simplify the expression
Now we simplify the expression:

\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]

Thus,

\[ f(x) = \ln\left(\left(\frac{1}{\cos x} + \frac{\sin x}{\cos x}\right) \cdot \cos x\right) = \ln(1 + \sin x) \]

### Step 5: Find the minimum value of \(f(x)\)
To find the minimum value of \(f(x)\) in the interval \(0 < x < \frac{\pi}{2}\), we can evaluate it at \(x = \frac{\pi}{4}\):

\[ f\left(\frac{\pi}{4}\right) = \ln(1 + \sin\left(\frac{\pi}{4}\right)) = \ln(1 + \frac{\sqrt{2}}{2}) = \ln\left(\frac{2 + \sqrt{2}}{2}\right) \]

This can be rewritten as:

\[ f\left(\frac{\pi}{4}\right) = \ln(2 + \sqrt{2}) - \ln(2) = \ln\left(\frac{2 + \sqrt{2}}{2}\right) \]

### Step 6: Identify \(a\) and \(b\)
From the problem, we have:

\[ f(x) = \ln(a + \sqrt{b}) \]

Comparing with our result, we see:

\[ a = 2, \quad b = 2 \]

### Step 7: Calculate \(a + b\)
Finally, we calculate:

\[ a + b = 2 + 2 = 4 \]

### Conclusion
Thus, the value of \(a + b\) is:

\[ \boxed{4} \]