`int _(0) ^(pi //4) x. Sec ^(2) x dx =?`

Options

A. `(pi)/(4) +log sqrt(2)`
B. `(pi)/(4) -log sqrt(2)`
C. `1+ log sqrt(2)`
D. `1-(1)/(2)log 2`

Answer

To solve the integral

\[ I = \int_0^{\frac{\pi}{4}} x \sec^2 x \, dx \]

we will use the method of integration by parts. The formula for integration by parts is:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)

Let:
- \( u = x \) (which means \( du = dx \))
- \( dv = \sec^2 x \, dx \) (which means \( v = \tan x \))

### Step 2: Apply the integration by parts formula

Now, applying the integration by parts formula:

\[ I = \left[ x \tan x \right]_0^{\frac{\pi}{4}} - \int_0^{\frac{\pi}{4}} \tan x \, dx \]

### Step 3: Evaluate the first term

Now we evaluate the first term:

\[ \left[ x \tan x \right]_0^{\frac{\pi}{4}} = \left( \frac{\pi}{4} \tan \frac{\pi}{4} \right) - \left( 0 \cdot \tan 0 \right) \]

Since \( \tan \frac{\pi}{4} = 1 \):

\[ = \frac{\pi}{4} \cdot 1 - 0 = \frac{\pi}{4} \]

### Step 4: Evaluate the integral \( \int \tan x \, dx \)

Now we need to evaluate \( \int \tan x \, dx \):

\[ \int \tan x \, dx = -\log |\cos x| + C \]

Thus, we need to evaluate:

\[ \int_0^{\frac{\pi}{4}} \tan x \, dx = \left[-\log |\cos x|\right]_0^{\frac{\pi}{4}} \]

### Step 5: Evaluate the limits for \( \tan x \)

Calculating the limits:

\[ = -\log |\cos \frac{\pi}{4}| + \log |\cos 0| \]

Since \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \) and \( \cos 0 = 1 \):

\[ = -\log \left(\frac{1}{\sqrt{2}}\right) + \log(1) \]

Since \( \log(1) = 0 \):

\[ = -\log \left(\frac{1}{\sqrt{2}}\right) = \log(\sqrt{2}) = \frac{1}{2} \log(2) \]

### Step 6: Combine the results

Now substituting back into our expression for \( I \):

\[ I = \frac{\pi}{4} - \left(-\log \left(\frac{1}{\sqrt{2}}\right)\right) = \frac{\pi}{4} - \log(\sqrt{2}) \]

### Step 7: Final answer

Thus, the final answer is:

\[ I = \frac{\pi}{4} - \log(\sqrt{2}) \]