`int e ^(xsin x +cos x )((x ^(4) cos ^(3) x-x sin x +cos x)/(x ^(2) cos ^(2) x )) dx =`

Options

A. `e^((x sin x + cos x))*(x+(1)/(x cos x))+C`
B. `e^((x sin x + cos x ))*(x cos x (1)/(x))+C`
C. `e^((x sin x + cos x))*(x- (1)/(x cos x))+C`
D. None of the above

Answer

To solve the integral

\[ I = \int e^{x \sin x + \cos x} \left( \frac{x^4 \cos^3 x - x \sin x + \cos x}{x^2 \cos^2 x} \right) dx, \]

we will break it down into manageable steps.

### Step 1: Simplify the Integral
We can rewrite the integral as:

\[ I = \int e^{x \sin x + \cos x} \left( \frac{x^4 \cos^3 x}{x^2 \cos^2 x} - \frac{x \sin x}{x^2 \cos^2 x} + \frac{\cos x}{x^2 \cos^2 x} \right) dx. \]

This simplifies to:

\[ I = \int e^{x \sin x + \cos x} \left( x^2 \cos x - \frac{\sin x}{x \cos^2 x} + \frac{1}{x^2 \cos x} \right) dx. \]

### Step 2: Break Down the Integral
Let’s denote:

\[ I_1 = \int e^{x \sin x + \cos x} \left( x^2 \cos x \right) dx, \]
\[ I_2 = \int e^{x \sin x + \cos x} \left( -\frac{\sin x}{x \cos^2 x} + \frac{1}{x^2 \cos x} \right) dx. \]

Thus, we can write:

\[ I = I_1 + I_2. \]

### Step 3: Solve \(I_1\) using Integration by Parts
For \(I_1\), we will use integration by parts. Let:

- \(u = e^{x \sin x + \cos x}\)
- \(dv = x^2 \cos x \, dx\)

Then, we differentiate \(u\) and integrate \(dv\):

- \(du = e^{x \sin x + \cos x} (x \cos x + \sin x) \, dx\)
- \(v = \frac{x^2 \sin x + 2x \cos x}{2}\)

Applying integration by parts:

\[ I_1 = u v - \int v \, du. \]

### Step 4: Solve \(I_2\)
For \(I_2\), we can simplify the integral further. We will need to handle each term separately:

1. For \(-\frac{\sin x}{x \cos^2 x}\), we can use substitution.
2. For \(\frac{1}{x^2 \cos x}\), we can also use substitution.

Let’s denote:

\[ t = x \cos x \implies dt = (\cos x - x \sin x) \, dx. \]

### Step 5: Combine Results
After evaluating both \(I_1\) and \(I_2\), we combine the results:

\[ I = I_1 + I_2. \]

### Final Result
The final result will be:

\[ I = x e^{x \sin x + \cos x} - e^{x \sin x + \cos x} \left( \frac{1}{x \cos x} \right) + C, \]

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

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