`int(e^x(1+x))/(cos^2 (e^x * x) )dx` equals

Options

A. (A) `-cot(e ^x*x)+C`
B. (B) `tan(x e^x)+C`
C. (C) `tan(e^x)+C`
D. (D) `cot(e^x)+C`

Explanation

To solve the integral \( \int \frac{e^x(1+x)}{\cos^2(e^x x)} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
The integral can be expressed as:
\[ \int \frac{e^x + e^x x}{\cos^2(e^x x)} \, dx \]

### Step 2: Use Substitution
Let \( t = e^x x \). We will differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = e^x x + e^x = e^x (x + 1) \]
This implies:
\[ dt = e^x (1 + x) \, dx \]
Thus, we can rewrite the integral in terms of \( t \):
\[ dx = \frac{dt}{e^x (1 + x)} \]
Substituting this into the integral gives:
\[ \int \frac{dt}{\cos^2(t)} \]

### Step 3: Simplify the Integral
The integral \( \int \frac{dt}{\cos^2(t)} \) is equivalent to:
\[ \int \sec^2(t) \, dt \]

### Step 4: Integrate
The integral of \( \sec^2(t) \) is:
\[ \tan(t) + C \]

### Step 5: Substitute Back
Now, substitute back \( t = e^x x \):
\[ \tan(e^x x) + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{e^x(1+x)}{\cos^2(e^x x)} \, dx = \tan(e^x x) + C \]

---