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 \( \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx \) and find the value of \( m \) such that \( \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx = \frac{\pi}{\sqrt{m}} \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx \]
We can rewrite \( \sqrt{\tan x} \) and \( \sqrt{\cot x} \) in terms of sine and cosine:
\[ \sqrt{\tan x} = \frac{\sqrt{\sin x}}{\sqrt{\cos x}}, \quad \sqrt{\cot x} = \frac{\sqrt{\cos x}}{\sqrt{\sin x}} \]
Thus, the integral becomes:
\[ \int_{0}^{\frac{\pi}{4}} \left( \frac{\sqrt{\sin x}}{\sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\sin x}} \right) dx \]

### Step 2: Combine the Terms
Next, we combine the two fractions:
\[ \int_{0}^{\frac{\pi}{4}} \left( \frac{\sqrt{\sin x} \cdot \sqrt{\sin x} + \sqrt{\cos x} \cdot \sqrt{\cos x}}{\sqrt{\sin x \cos x}} \right) dx = \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{\sqrt{\sin x \cos x}} dx \]

### Step 3: Simplify the Denominator
Using the identity \( \sin 2x = 2 \sin x \cos x \), we can rewrite the integral:
\[ \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{\sqrt{\frac{1}{2} \sin 2x}} dx = \sqrt{2} \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx \]

### Step 4: Change of Variables
Let \( t = \sin x - \cos x \). Then, we differentiate:
\[ dt = (\cos x + \sin x) dx \]
The limits change as follows:
- When \( x = 0 \), \( t = -1 \)
- When \( x = \frac{\pi}{4} \), \( t = 0 \)

Thus, we can rewrite the integral:
\[ \sqrt{2} \int_{-1}^{0} \frac{1}{\sqrt{1 - t^2}} dt \]

### Step 5: Evaluate the Integral
The integral \( \int \frac{1}{\sqrt{1 - t^2}} dt \) is known to be \( \sin^{-1}(t) \):
\[ \sqrt{2} \left[ \sin^{-1}(t) \right]_{-1}^{0} = \sqrt{2} \left( \sin^{-1}(0) - \sin^{-1}(-1) \right) = \sqrt{2} \left( 0 - \left(-\frac{\pi}{2}\right) \right) = \frac{\pi \sqrt{2}}{2} \]

### Step 6: Relate to Given Equation
Now we have:
\[ \int_{0}^{\frac{\pi}{4}} \left( \sqrt{\tan x} + \sqrt{\cot x} \right) dx = \frac{\pi \sqrt{2}}{2} \]
We compare this with the given equation:
\[ \frac{\pi}{\sqrt{m}} = \frac{\pi \sqrt{2}}{2} \]

### Step 7: Solve for \( m \)
Equating the two expressions gives:
\[ \sqrt{m} = 2 \sqrt{2} \implies m = (2 \sqrt{2})^2 = 8 \]

Thus, the value of \( m \) is:
\[ \boxed{8} \]