If `int (dx)/(5+4 cos x) = k tan^(-1) (m tan ((x)/(2))) + C` then :

Options

A. ` k = (1)/(3), m = (1)/(3)`
B. `k = (1)/(3), m = (2)/(3)`
C. `k = (2)/(3), m = (2)/(3)`
D. `k = (2)/(3), = (2)/(3)`

Answer

To solve the integral

\[ I = \int \frac{dx}{5 + 4 \cos x} \]

we will follow these steps:

### Step 1: Rewrite \(\cos x\) in terms of \(\tan\)
We can use the half-angle identity for cosine:
\[ \cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]
Let \( t = \tan\left(\frac{x}{2}\right) \). Then, we can rewrite the integral as:
\[ I = \int \frac{dx}{5 + 4 \left(\frac{1 - t^2}{1 + t^2}\right)} \]

### Step 2: Substitute and simplify the integral
Substituting \(\cos x\) into the integral gives:
\[ I = \int \frac{dx}{5 + \frac{4(1 - t^2)}{1 + t^2}} = \int \frac{(1 + t^2) dx}{5(1 + t^2) + 4(1 - t^2)} \]
Simplifying the denominator:
\[ 5(1 + t^2) + 4(1 - t^2) = 5 + 5t^2 + 4 - 4t^2 = 9 + t^2 \]
Thus, we have:
\[ I = \int \frac{(1 + t^2) dx}{9 + t^2} \]

### Step 3: Change of variable
Now, we need to express \(dx\) in terms of \(dt\). From the substitution \(t = \tan\left(\frac{x}{2}\right)\), we differentiate:
\[ dx = \frac{2}{1 + t^2} dt \]
Substituting this into the integral:
\[ I = \int \frac{(1 + t^2) \cdot \frac{2}{1 + t^2} dt}{9 + t^2} = 2 \int \frac{dt}{9 + t^2} \]

### Step 4: Solve the integral
The integral \(\int \frac{dt}{a^2 + t^2}\) is known to be:
\[ \frac{1}{a} \tan^{-1}\left(\frac{t}{a}\right) + C \]
In our case, \(a^2 = 9\) so \(a = 3\):
\[ I = 2 \cdot \frac{1}{3} \tan^{-1}\left(\frac{t}{3}\right) + C = \frac{2}{3} \tan^{-1}\left(\frac{t}{3}\right) + C \]

### Step 5: Substitute back for \(t\)
Recall that \(t = \tan\left(\frac{x}{2}\right)\):
\[ I = \frac{2}{3} \tan^{-1}\left(\frac{\tan\left(\frac{x}{2}\right)}{3}\right) + C \]

### Step 6: Compare with the given form
We need to compare this with the given form:
\[ I = k \tan^{-1}(m \tan\left(\frac{x}{2}\right)) + C \]
From our result, we see:
\[ k = \frac{2}{3} \quad \text{and} \quad m = \frac{1}{3} \]

### Conclusion
Thus, the values of \(k\) and \(m\) are:
\[ k = \frac{2}{3}, \quad m = \frac{1}{3} \]

### Final Answer
The correct option is **C: \(k = \frac{2}{3}\) and \(m = \frac{1}{3}\)**.

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