If the integal `int (5 tan x dx )/(tan x-2) =x +a ln |sin x-2 cos x | +C,` then 'a' is equal to :

Options

A. 1
B. 2
C. `-1`
D. `-2`

Answer

To solve the integral

\[ \int \frac{5 \tan x \, dx}{\tan x - 2} \]

and find the value of \( a \) in the expression

\[ \int \frac{5 \tan x \, dx}{\tan x - 2} = x + a \ln |\sin x - 2 \cos x| + C, \]

we will follow these steps:

### Step 1: Rewrite the integral
We can rewrite \( \tan x \) in terms of sine and cosine:

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

Thus, the integral becomes:

\[ \int \frac{5 \frac{\sin x}{\cos x} \, dx}{\frac{\sin x}{\cos x} - 2} = \int \frac{5 \sin x \, dx}{\sin x - 2 \cos x} \]

### Step 2: Use Integration by Parts
We will use the method of partial fractions. We can express \( \frac{5 \sin x}{\sin x - 2 \cos x} \) as:

\[ \frac{5 \sin x}{\sin x - 2 \cos x} = \frac{A}{\sin x - 2 \cos x} + B \cdot \frac{\cos x + 2 \sin x}{\sin x - 2 \cos x} \]

### Step 3: Set up the equations
To find \( A \) and \( B \), we multiply both sides by the denominator \( \sin x - 2 \cos x \):

\[ 5 \sin x = A + B (\cos x + 2 \sin x) \]

Expanding this gives:

\[ 5 \sin x = A + B \cos x + 2B \sin x \]

### Step 4: Compare coefficients
Now we can compare coefficients for \( \sin x \) and \( \cos x \):

1. Coefficient of \( \sin x \): \( 5 = 2B \)
2. Coefficient of \( \cos x \): \( 0 = B \)

From the second equation, we find \( B = 0 \). Substituting \( B = 0 \) into the first equation gives:

\[ 5 = 2(0) \implies A = 5 \]

### Step 5: Integrate
Now we can integrate:

\[ \int \frac{5 \sin x}{\sin x - 2 \cos x} \, dx = 5 \int \frac{1}{\sin x - 2 \cos x} \, dx \]

Let \( t = \sin x - 2 \cos x \). Then, the derivative \( dt = \cos x + 2 \sin x \, dx \).

### Step 6: Substitute and integrate
We can express the integral in terms of \( t \):

\[ \int \frac{5 \sin x}{t} \, dt \]

This leads to:

\[ = 5 \ln |t| + C = 5 \ln |\sin x - 2 \cos x| + C \]

### Step 7: Compare with the given expression
Now we compare:

\[ x + a \ln |\sin x - 2 \cos x| + C \]

From this, we see that \( a = 5 \).

### Final Answer
Thus, the value of \( a \) is:

\[ \boxed{5} \]