If `int_(0)^(pi//4) (x sin x)/( cos^(3) x) dx= (pi)/(4) +A`, then A is equal to

Options

A. `0`
B. `1/2`
C. `1`
D. `-1/2`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{4}} \frac{x \sin x}{\cos^3 x} \, dx \) and find the value of \( A \) such that \( I = \frac{\pi}{4} + A \), we will use integration by parts.

### Step 1: Set up integration by parts
We will use the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Let:
- \( u = x \) (thus \( du = dx \))
- \( dv = \frac{\sin x}{\cos^3 x} \, dx \)

### Step 2: Find \( v \)
To find \( v \), we need to integrate \( dv \):
\[ v = \int \frac{\sin x}{\cos^3 x} \, dx \]
Using the substitution \( w = \cos x \), we have \( dw = -\sin x \, dx \), or \( \sin x \, dx = -dw \). Thus, we can rewrite the integral:
\[ v = -\int \frac{1}{w^3} \, dw = \frac{1}{2w^2} = \frac{1}{2 \cos^2 x} \]

### Step 3: Apply integration by parts
Now we apply the integration by parts formula:
\[ I = uv \bigg|_{0}^{\frac{\pi}{4}} - \int v \, du \]
Substituting \( u \) and \( v \):
\[ I = \left[ x \cdot \frac{1}{2 \cos^2 x} \right]_{0}^{\frac{\pi}{4}} - \int_{0}^{\frac{\pi}{4}} \frac{1}{2 \cos^2 x} \, dx \]

### Step 4: Evaluate the boundary term
Now we evaluate the boundary term:
\[ \left[ x \cdot \frac{1}{2 \cos^2 x} \right]_{0}^{\frac{\pi}{4}} = \left( \frac{\pi/4}{2 \cdot \cos^2(\pi/4)} \right) - \left( 0 \cdot \frac{1}{2 \cos^2(0)} \right) \]
Calculating \( \cos^2(\pi/4) = \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2} \):
\[ = \frac{\pi/4}{2 \cdot \frac{1}{2}} = \frac{\pi/4}{1} = \frac{\pi}{4} \]

### Step 5: Evaluate the integral term
Now we need to evaluate the integral:
\[ \int_{0}^{\frac{\pi}{4}} \frac{1}{2 \cos^2 x} \, dx = \frac{1}{2} \int_{0}^{\frac{\pi}{4}} \sec^2 x \, dx \]
The integral of \( \sec^2 x \) is \( \tan x \):
\[ = \frac{1}{2} \left[ \tan x \right]_{0}^{\frac{\pi}{4}} = \frac{1}{2} \left( \tan\left(\frac{\pi}{4}\right) - \tan(0) \right) = \frac{1}{2} (1 - 0) = \frac{1}{2} \]

### Step 6: Combine results
Putting it all together:
\[ I = \frac{\pi}{4} - \frac{1}{2} \]
Thus, we have:
\[ I = \frac{\pi}{4} - \frac{1}{2} \]
Given \( I = \frac{\pi}{4} + A \), we can equate:
\[ \frac{\pi}{4} - \frac{1}{2} = \frac{\pi}{4} + A \]
This leads to:
\[ A = -\frac{1}{2} \]

### Final Answer
Thus, the value of \( A \) is:
\[ \boxed{-\frac{1}{2}} \]