`int_(0)^(pi/4)(x.sinx)/(cos^(3)x)` dx equal to :

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{4}} \frac{x \sin x}{\cos^3 x} \, dx \), we can use integration by parts.

### Step-by-Step Solution:

1. **Identify the parts for integration by parts**:
Let \( u = x \) and \( dv = \frac{\sin x}{\cos^3 x} \, dx \).
Then, we differentiate and integrate respectively:
\[ du = dx \quad \text{and} \quad v = \int \frac{\sin x}{\cos^3 x} \, dx. \]

2. **Evaluate \( v \)**:
To find \( v \), we can use the substitution \( t = \cos x \). Then, \( dt = -\sin x \, dx \) or \( \sin x \, dx = -dt \). The limits change as follows:
- When \( x = 0 \), \( t = \cos(0) = 1 \).
- When \( x = \frac{\pi}{4} \), \( t = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \).

Thus, we have:
\[ v = \int \frac{\sin x}{\cos^3 x} \, dx = \int \frac{-dt}{t^3} = \int -t^{-3} \, dt = \frac{1}{2} t^{-2} + C = \frac{1}{2 \cos^2 x} + C. \]

3. **Apply integration by parts**:
Using the integration by parts formula \( I = uv - \int v \, du \):
\[ I = x \cdot \frac{1}{2 \cos^2 x} \bigg|_{0}^{\frac{\pi}{4}} - \int \frac{1}{2 \cos^2 x} \, dx. \]

4. **Evaluate the first term**:
Calculate \( x \cdot \frac{1}{2 \cos^2 x} \) at the limits:
- At \( x = \frac{\pi}{4} \):
\[ \frac{\pi}{4} \cdot \frac{1}{2 \left(\frac{1}{\sqrt{2}}\right)^2} = \frac{\pi}{4} \cdot \frac{1}{2 \cdot \frac{1}{2}} = \frac{\pi}{4} \cdot 1 = \frac{\pi}{4}. \]
- At \( x = 0 \):
\[ 0 \cdot \frac{1}{2 \cos^2(0)} = 0. \]
Therefore, the first term evaluates to \( \frac{\pi}{4} - 0 = \frac{\pi}{4} \).

5. **Evaluate the second integral**:
Now we need to evaluate \( \int \frac{1}{2 \cos^2 x} \, dx \):
\[ \int \frac{1}{2 \cos^2 x} \, dx = \frac{1}{2} \int \sec^2 x \, dx = \frac{1}{2} \tan x \bigg|_{0}^{\frac{\pi}{4}}. \]
Evaluating this gives:
- At \( x = \frac{\pi}{4} \):
\[ \tan\left(\frac{\pi}{4}\right) = 1. \]
- At \( x = 0 \):
\[ \tan(0) = 0. \]
Thus, this evaluates to \( \frac{1}{2} (1 - 0) = \frac{1}{2} \).

6. **Combine results**:
Putting it all together:
\[ I = \frac{\pi}{4} - \frac{1}{2}. \]

### Final Answer:
\[ I = \frac{\pi}{4} - \frac{1}{2}. \]