`int e^(tan x) (sec^(2) x + sec^(3) x sin x ) dx` is equal to

Options

A. `sec x e^(tanx ) + c`
B. `tan x e^(tan x) + c`
C. `e^(tan x ) + tan x + c`
D. `(1 + tan x ) e^(tan x) + c`

Explanation

To solve the integral \( I = \int e^{\tan x} \left( \sec^2 x + \sec^3 x \sin x \right) dx \), we can follow these steps:

### Step 1: Simplify the integrand
We can factor out \( \sec^2 x \) from the integrand:
\[ I = \int e^{\tan x} \sec^2 x \left( 1 + \sec x \sin x \right) dx \]

### Step 2: Rewrite \( \sec x \) and \( \sin x \)
Recall that:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \sin x = \frac{\tan x}{\sec x} \]
Thus, we can express \( \sec x \sin x \) as:
\[ \sec x \sin x = \frac{\sin x}{\cos x} = \tan x \]
So, we can rewrite the integral as:
\[ I = \int e^{\tan x} \sec^2 x (1 + \tan x) dx \]

### Step 3: Substitution
Let \( t = \tan x \). Then, the derivative \( dt = \sec^2 x \, dx \). Therefore, we can rewrite the integral in terms of \( t \):
\[ I = \int e^t (1 + t) dt \]

### Step 4: Split the integral
Now we can split the integral:
\[ I = \int e^t dt + \int t e^t dt \]

### Step 5: Solve the first integral
The integral of \( e^t \) is:
\[ \int e^t dt = e^t \]

### Step 6: Solve the second integral using integration by parts
For the integral \( \int t e^t dt \), we use integration by parts:
Let \( u = t \) and \( dv = e^t dt \).
Then, \( du = dt \) and \( v = e^t \).

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
we get:
\[ \int t e^t dt = t e^t - \int e^t dt = t e^t - e^t \]

### Step 7: Combine results
Now we combine the results:
\[ I = e^t + (t e^t - e^t) = t e^t \]

### Step 8: Substitute back for \( t \)
Recall that \( t = \tan x \):
\[ I = \tan x \cdot e^{\tan x} \]

### Step 9: Add the constant of integration
Finally, we add the constant of integration \( C \):
\[ I = \tan x \cdot e^{\tan x} + C \]

### Final Answer
Thus, the integral \( \int e^{\tan x} \left( \sec^2 x + \sec^3 x \sin x \right) dx \) is equal to:
\[ \tan x \cdot e^{\tan x} + C \]