`int_(0)^(pi/3) tantheta/(sqrt(2ksectheta))d theta=1-1/sqrt2,(kgt0)`, then the value of k is

Options

A. 2
B. `1/2`
C. 4
D. 1

Answer

To solve the integral equation given in the problem, we start with the expression:

\[ \int_{0}^{\frac{\pi}{3}} \frac{\tan \theta}{\sqrt{2k \sec \theta}} d\theta = 1 - \frac{1}{\sqrt{2}} \]

### Step 1: Simplify the Integral

We can rewrite the integral as:

\[ \int_{0}^{\frac{\pi}{3}} \frac{\tan \theta}{\sqrt{2k} \sec \theta}} d\theta = \frac{1}{\sqrt{2k}} \int_{0}^{\frac{\pi}{3}} \tan \theta \cos \theta d\theta \]

Since \(\sec \theta = \frac{1}{\cos \theta}\), we have:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

Thus, the integral becomes:

\[ \int_{0}^{\frac{\pi}{3}} \frac{\sin \theta}{\sqrt{2k}} d\theta \]

### Step 2: Change of Variable

To evaluate the integral, we can use the substitution \(u = \cos \theta\). Then, \(du = -\sin \theta d\theta\). The limits change as follows:
- When \(\theta = 0\), \(u = \cos(0) = 1\)
- When \(\theta = \frac{\pi}{3}\), \(u = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)

Thus, the integral becomes:

\[ -\frac{1}{\sqrt{2k}} \int_{1}^{\frac{1}{2}} du \]

### Step 3: Evaluate the Integral

Now we can evaluate the integral:

\[ -\frac{1}{\sqrt{2k}} \left[ u \right]_{1}^{\frac{1}{2}} = -\frac{1}{\sqrt{2k}} \left( \frac{1}{2} - 1 \right) = -\frac{1}{\sqrt{2k}} \left( -\frac{1}{2} \right) = \frac{1}{2\sqrt{2k}} \]

### Step 4: Set Up the Equation

Now we equate this to the right-hand side of the original equation:

\[ \frac{1}{2\sqrt{2k}} = 1 - \frac{1}{\sqrt{2}} \]

### Step 5: Solve for \(k\)

To find \(k\), we first simplify the right-hand side:

\[ 1 - \frac{1}{\sqrt{2}} = \frac{\sqrt{2} - 1}{\sqrt{2}} \]

Now, equate the two expressions:

\[ \frac{1}{2\sqrt{2k}} = \frac{\sqrt{2} - 1}{\sqrt{2}} \]

Cross-multiplying gives:

\[ \sqrt{2} = 2\sqrt{2k}(\sqrt{2} - 1) \]

Squaring both sides:

\[ 2 = 4(2k)(\sqrt{2} - 1)^2 \]

Now, solving for \(k\):

\[ k = \frac{2}{4(2)(\sqrt{2} - 1)^2} = \frac{1}{4(\sqrt{2} - 1)^2} \]

### Final Value of \(k\)

Thus, the value of \(k\) is:

\[ k = \frac{1}{4(\sqrt{2} - 1)^2} \]