If f(x) = sin x then `int(e^(f(x))(xcos^(3)x-f(x)))/(1-(f(x))^(2))dx` is equal to

Options

A. `e^(f(x))(x-sinx)+C`
B. `e^(f(x))(x-secx)+C`
C. `e^(f(x))(x+secx)+C`
D. `e^(f(x))(x+sec^(2))+C`

Answer

To solve the integral

\[ I = \int \frac{e^{f(x)}(x \cos^3 x - f(x))}{1 - (f(x))^2} \, dx \]

where \( f(x) = \sin x \), we will follow these steps:

### Step 1: Substitute \( f(x) \)
First, we substitute \( f(x) = \sin x \) into the integral:

\[ I = \int \frac{e^{\sin x}(x \cos^3 x - \sin x)}{1 - \sin^2 x} \, dx \]

### Step 2: Simplify the Denominator
Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), we can simplify the denominator:

\[ 1 - \sin^2 x = \cos^2 x \]

Thus, the integral becomes:

\[ I = \int \frac{e^{\sin x}(x \cos^3 x - \sin x)}{\cos^2 x} \, dx \]

### Step 3: Split the Integral
We can split the integral into two parts:

\[ I = \int e^{\sin x} \left( \frac{x \cos^3 x}{\cos^2 x} - \frac{\sin x}{\cos^2 x} \right) \, dx \]

This simplifies to:

\[ I = \int e^{\sin x} (x \cos x - \tan x) \, dx \]

### Step 4: Integration by Parts
Now we will use integration by parts. Let:

- \( u = e^{\sin x} \) and \( dv = (x \cos x - \tan x) \, dx \)

Then we need to find \( du \) and \( v \):

\[ du = e^{\sin x} \cos x \, dx \]

To find \( v \), we need to integrate \( (x \cos x - \tan x) \):

1. Integrate \( x \cos x \) using integration by parts again.
2. The integral of \( \tan x \) is \( -\ln |\cos x| \).

### Step 5: Combine Results
After performing the integration by parts, we will have:

\[ I = e^{\sin x} \left( \text{result from } v \right) - \int \left( \text{result from } du \right) \]

### Step 6: Final Expression
After simplifying and combining the results, we will arrive at:

\[ I = e^{\sin x} \left( x - \sec x \right) + C \]

where \( C \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:

\[ I = e^{f(x)} \left( x - \sec x \right) + C \]