The value of `int(xtanxsecx)/((tanx-x)^(2))dx` is equal to

Options

A. `(1)/(sinx-xcosx)+c`
B. `(1)/(xcosx-sinx)+c`
C. `(1)/(xsinx-cosx)+c`
D. none of these

Answer

To solve the integral \( I = \int \frac{x \tan x \sec x}{(\tan x - x)^2} \, dx \), we will follow a series of steps to simplify and evaluate the integral.

### Step 1: Rewrite the integral
We start by rewriting the integral in terms of sine and cosine:
\[ I = \int \frac{x \tan x \sec x}{(\tan x - x)^2} \, dx = \int \frac{x \frac{\sin x}{\cos x} \frac{1}{\cos x}}{(\frac{\sin x}{\cos x} - x)^2} \, dx \]
This simplifies to:
\[ I = \int \frac{x \sin x}{\cos^2 x \left(\frac{\sin x - x \cos x}{\cos x}\right)^2} \, dx \]

### Step 2: Simplify the expression
The expression inside the integral can be simplified further:
\[ I = \int \frac{x \sin x \cos^2 x}{(\sin x - x \cos x)^2} \, dx \]

### Step 3: Use substitution
Let \( v = \sin x - x \cos x \). Then, we differentiate \( v \):
\[ dv = (\cos x - x \sin x) \, dx \]
This means we can express \( dx \) in terms of \( dv \):
\[ dx = \frac{dv}{\cos x - x \sin x} \]

### Step 4: Substitute back into the integral
Now we substitute \( v \) and \( dx \) into the integral:
\[ I = \int \frac{x \sin x}{v^2} \cdot \frac{dv}{\cos x - x \sin x} \]

### Step 5: Recognize the structure
Notice that the numerator \( x \sin x \) is the derivative of \( v \) (up to a constant factor). This allows us to rewrite the integral:
\[ I = \int \frac{dv}{v^2} \]

### Step 6: Solve the integral
The integral \( \int \frac{dv}{v^2} \) can be solved using the power rule:
\[ I = -\frac{1}{v} + C \]

### Step 7: Substitute back for \( v \)
Now we substitute back for \( v \):
\[ I = -\frac{1}{\sin x - x \cos x} + C \]

### Final Answer
Thus, the value of the integral is:
\[ I = -\frac{1}{\sin x - x \cos x} + C \]