If `int_(0)^(pi//4)[sqrt(tanx)+sqrt(cotx)]dx=(pi)/(sqrtm),` then the value of m is equal to

Answer

To solve the integral

\[ \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx = \frac{\pi}{\sqrt{m}}, \]

we will follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in terms of sine and cosine:

\[ \sqrt{\tan x} = \sqrt{\frac{\sin x}{\cos x}} \quad \text{and} \quad \sqrt{\cot x} = \sqrt{\frac{\cos x}{\sin x}}. \]

Thus, we have:

\[ \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx = \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\frac{\sin x}{\cos x}} + \sqrt{\frac{\cos x}{\sin x}} \right) dx. \]

### Step 2: Combine the Terms
We can combine the two terms under a common denominator:

\[ \sqrt{\tan x} + \sqrt{\cot x} = \frac{\sqrt{\sin x \cos x} + \sqrt{\cos x \sin x}}{\sqrt{\sin x \cos x}} = \frac{2\sqrt{\sin x \cos x}}{\sqrt{\sin x \cos x}} = \frac{2}{\sqrt{\sin x \cos x}}. \]

### Step 3: Change of Variables
Now, we can simplify our integral:

\[ \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx = 2 \int_{0}^{\frac{\pi}{4}} \frac{1}{\sqrt{\sin x \cos x}} dx. \]

### Step 4: Use the Identity
We know that

\[ \sin 2x = 2 \sin x \cos x \implies \sin x \cos x = \frac{1}{2} \sin 2x. \]

Thus, we can rewrite the integral as:

\[ 2 \int_{0}^{\frac{\pi}{4}} \frac{1}{\sqrt{\frac{1}{2} \sin 2x}} dx = 2 \sqrt{2} \int_{0}^{\frac{\pi}{4}} \frac{1}{\sqrt{\sin 2x}} dx. \]

### Step 5: Change of Variables Again
Let \( t = \sin 2x \). Then, \( dt = 2 \cos 2x \, dx \) or \( dx = \frac{dt}{2\sqrt{1-t^2}} \). The limits change from \( x = 0 \) to \( x = \frac{\pi}{4} \) which corresponds to \( t = 0 \) to \( t = 1 \).

### Step 6: Evaluate the Integral
Now, we can evaluate the integral:

\[ 2 \sqrt{2} \int_{0}^{1} \frac{1}{\sqrt{t}} \cdot \frac{dt}{2\sqrt{1-t^2}} = \sqrt{2} \int_{0}^{1} \frac{1}{\sqrt{t(1-t^2)}} dt. \]

This integral is known and evaluates to \( \frac{\pi}{2} \).

### Step 7: Final Calculation
Thus, we have:

\[ \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx = \sqrt{2} \cdot \frac{\pi}{2} = \frac{\pi \sqrt{2}}{2}. \]

### Step 8: Equate and Solve for m
From the original equation, we compare:

\[ \frac{\pi \sqrt{2}}{2} = \frac{\pi}{\sqrt{m}}. \]

Cross-multiplying gives:

\[ \sqrt{m} = 2\sqrt{2} \implies m = 8. \]

### Conclusion
The value of \( m \) is

\[ \boxed{8}. \]