The value of `inte^(secx)*sec^3x(sin^2x+cosx+sinx+sinxcosx)dx` is

Options

A. `e^(secx)(sec^(2)+secxtanx)+C`
B. `e^(secx)+C`
C. `e^(secx)(secx+tanx)+C`
D. None of these

Answer

To solve the integral

\[ \int e^{\sec x} \sec^3 x \left( \sin^2 x + \cos x + \sin x + \sin x \cos x \right) dx, \]

we can follow these steps:

### Step 1: Identify the Integral
We start with the integral:

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

### Step 2: Simplify the Expression Inside the Integral
We can rearrange the terms inside the integral:

\[ \sin^2 x + \sin x \cos x + \sin x + \cos x = \sin^2 x + \sin x (\cos x + 1) + \cos x. \]

### Step 3: Differentiate the Proposed Solution
We will check the options given in the problem. Let's consider option C:

\[ e^{\sec x} \left( \sec x + \tan x \right) + C. \]

To verify, we differentiate this expression using the product rule. Let

\[ u = e^{\sec x}, \quad v = \sec x + \tan x. \]

Then,

\[ \frac{du}{dx} = e^{\sec x} \sec x \tan x, \]

and

\[ \frac{dv}{dx} = \sec x \tan x + \sec^2 x. \]

Using the product rule:

\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}. \]

Substituting the derivatives we found:

\[ \frac{d}{dx}(e^{\sec x} (\sec x + \tan x)) = e^{\sec x} \left( \sec x \tan x + \sec^2 x \right) + (\sec x + \tan x) e^{\sec x} \sec x \tan x. \]

### Step 4: Factor Out \( e^{\sec x} \)
Factoring out \( e^{\sec x} \):

\[ = e^{\sec x} \left( \sec x \tan x + \sec^2 x + (\sec x + \tan x) \sec x \tan x \right). \]

### Step 5: Combine Like Terms
Now we combine the terms:

\[ = e^{\sec x} \left( \sec^2 x + \sec x \tan x + \sec^2 x \tan x + \tan^2 x \right). \]

### Step 6: Recognize the Original Integral
This expression simplifies to:

\[ = e^{\sec x} \sec^3 x \left( \sin^2 x + \cos x + \sin x + \sin x \cos x \right), \]

which matches the original integral we started with.

### Step 7: Conclusion
Thus, we conclude that the value of the integral is:

\[ \int e^{\sec x} \sec^3 x \left( \sin^2 x + \cos x + \sin x + \sin x \cos x \right) dx = e^{\sec x} \left( \sec x + \tan x \right) + C. \]