What is the value of `int_(pi//6)^(pi//4) (dx)/(sin x cos x)`

Options

A. A. `2 log sqrt3`
B. B. `log sqrt3`
C. C. `2log 3`
D. D. `4 log 3`

Answer

To solve the integral \( \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{dx}{\sin x \cos x} \), we can follow these steps:

### Step 1: Rewrite the integrand
We start by rewriting the integrand using the identity \( \sin x \cos x = \frac{1}{2} \sin(2x) \):
\[ \frac{1}{\sin x \cos x} = \frac{2}{\sin(2x)} \]
Thus, the integral becomes:
\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{dx}{\sin x \cos x} = 2 \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{dx}{\sin(2x)} \]

### Step 2: Change of variable
Next, we can use the substitution \( u = 2x \), which gives \( du = 2dx \) or \( dx = \frac{du}{2} \). We also need to change the limits of integration:
- When \( x = \frac{\pi}{6} \), \( u = \frac{\pi}{3} \)
- When \( x = \frac{\pi}{4} \), \( u = \frac{\pi}{2} \)

Substituting these into the integral, we have:
\[ 2 \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{dx}{\sin(2x)} = 2 \cdot \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{du}{\sin u} = \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{du}{\sin u} \]

### Step 3: Evaluate the integral
The integral \( \int \frac{du}{\sin u} \) can be evaluated as:
\[ \int \frac{du}{\sin u} = \log \left| \tan \left( \frac{u}{2} \right) \right| + C \]
Thus, we evaluate:
\[ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{du}{\sin u} = \left[ \log \left| \tan \left( \frac{u}{2} \right) \right| \right]_{\frac{\pi}{3}}^{\frac{\pi}{2}} \]

### Step 4: Apply the limits
Now we apply the limits:
1. For \( u = \frac{\pi}{2} \):
\[ \tan \left( \frac{\pi/2}{2} \right) = \tan \left( \frac{\pi}{4} \right) = 1 \implies \log(1) = 0 \]
2. For \( u = \frac{\pi}{3} \):
\[ \tan \left( \frac{\pi/3}{2} \right) = \tan \left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}} \implies \log \left( \frac{1}{\sqrt{3}} \right) = -\frac{1}{2} \log(3) \]

Putting it all together:
\[ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{du}{\sin u} = 0 - \left(-\frac{1}{2} \log(3)\right) = \frac{1}{2} \log(3) \]

### Final Answer
Thus, the value of the integral \( \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{dx}{\sin x \cos x} \) is:
\[ \frac{1}{2} \log(3) \]