`int_(pi//4) ^(pi//2) (cos theta)/((cos (theta/2) + sin (theta/2))^(3))d theta=`

Options

A. `2/(cos(pi/8)+sin(pi/8))+sqrt(2)`
B. `(-2)/(cos(pi/8)+sin(pi/8))+sqrt(2)`
C. `(2)/(cos(pi/8)+sin(pi/8))-sqrt(2)`
D. `(-2)/(cos(pi/8)+sin(pi/8))-sqrt(2)`

Answer

To solve the integral

\[ I = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos \theta}{(\cos \frac{\theta}{2} + \sin \frac{\theta}{2})^3} \, d\theta, \]

we will follow these steps:

### Step 1: Rewrite \(\cos \theta\)
We can express \(\cos \theta\) in terms of \(\cos \frac{\theta}{2}\) and \(\sin \frac{\theta}{2}\):
\[ \cos \theta = \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2}. \]

### Step 2: Substitute in the integral
Substituting this into the integral gives:
\[ I = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2}}{(\cos \frac{\theta}{2} + \sin \frac{\theta}{2})^3} \, d\theta. \]

### Step 3: Factor the numerator
We can factor the numerator using the identity \(a^2 - b^2 = (a-b)(a+b)\):
\[ I = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(\cos \frac{\theta}{2} - \sin \frac{\theta}{2})(\cos \frac{\theta}{2} + \sin \frac{\theta}{2})}{(\cos \frac{\theta}{2} + \sin \frac{\theta}{2})^3} \, d\theta. \]
This simplifies to:
\[ I = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}}{(\cos \frac{\theta}{2} + \sin \frac{\theta}{2})^2} \, d\theta. \]

### Step 4: Make a substitution
Let \(t = \cos \frac{\theta}{2} + \sin \frac{\theta}{2}\). Then we need to find \(d\theta\) in terms of \(dt\):
\[ \frac{d\theta}{d\left(\frac{\theta}{2}\right)} = 2 \implies d\theta = 2 \, d\left(\frac{\theta}{2}\right). \]
Now we can differentiate \(t\):
\[ dt = \left(-\frac{1}{2} \sin \frac{\theta}{2} + \frac{1}{2} \cos \frac{\theta}{2}\right) d\theta. \]

### Step 5: Change the limits
When \(\theta = \frac{\pi}{4}\):
\[ t = \cos \frac{\pi}{8} + \sin \frac{\pi}{8}, \]
and when \(\theta = \frac{\pi}{2}\):
\[ t = \cos \frac{\pi}{4} + \sin \frac{\pi}{4} = \sqrt{2}. \]

### Step 6: Substitute back into the integral
Now, substituting everything back into the integral:
\[ I = \int_{t(\frac{\pi}{4})}^{t(\frac{\pi}{2})} \frac{(\cos \frac{\theta}{2} - \sin \frac{\theta}{2})}{t^2} \cdot 2 \, dt. \]

### Step 7: Evaluate the integral
The integral simplifies to:
\[ I = 2 \int_{t(\frac{\pi}{4})}^{\sqrt{2}} (t - 1) t^{-2} \, dt. \]
This can be integrated to:
\[ I = 2 \left[-\frac{1}{t}\right]_{t(\frac{\pi}{4})}^{\sqrt{2}}. \]

### Step 8: Substitute the limits
Substituting the limits gives:
\[ I = 2 \left(-\frac{1}{\sqrt{2}} + \frac{1}{\cos \frac{\pi}{8} + \sin \frac{\pi}{8}}\right). \]

### Step 9: Final simplification
This results in:
\[ I = -2 \left(\frac{1}{\sqrt{2}} - \frac{1}{\cos \frac{\pi}{8} + \sin \frac{\pi}{8}}\right). \]

### Final Answer
The final answer is:
\[ I = -2 \left(\frac{1}{\sqrt{2}} - \frac{1}{\cos \frac{\pi}{8} + \sin \frac{\pi}{8}}\right). \]