`inte^(x)(1+x)sec^(2)(xe^(x))dx=f (x)+` Constant , then f (x) is equal to

Options

A. `cos(xe^(x))`
B. `sin(xe^(x))`
C. `2tan^(-1)x`
D. `tan(xe^(x))`

Explanation

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

### Step 1: Substitution
Let \( t = x e^x \). Then, we need to find \( dt \).

### Step 2: Differentiate \( t \)
Using the product rule:
\[ dt = (e^x + x e^x) \, dx = e^x (1 + x) \, dx \]
This means:
\[ dx = \frac{dt}{e^x (1 + x)} \]

### Step 3: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int e^x (1 + x) \sec^2(x e^x) \, dx = \int \sec^2(t) \, dt \]

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

### Step 5: Substitute Back
Now, substituting back \( t = x e^x \):
\[ \tan(t) = \tan(x e^x) \]
Thus, we have:
\[ \int e^x (1 + x) \sec^2(x e^x) \, dx = \tan(x e^x) + C \]

### Conclusion
Therefore, \( f(x) = \tan(x e^x) \).