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

Options

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

Answer

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

### Step 1: Substitution
Let \( t = x e^x \). We will differentiate this with respect to \( x \):

\[ \frac{dt}{dx} = e^x + x e^x = e^x(1+x) \]

Thus, we can express \( dt \) as:

\[ dt = e^x(1+x) \, dx \]

### Step 2: Rewrite the Integral
Now, we can rewrite the integral in terms of \( t \):

\[ \int e^x (1+x) \sec^2(x e^x) \, dx = \int \sec^2(t) \, dt \]

### Step 3: Integrate
The integral of \( \sec^2(t) \) is:

\[ \int \sec^2(t) \, dt = \tan(t) + C \]

### Step 4: Back Substitute
Now we substitute 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
From the above steps, we can conclude that:

\[ f(x) = \tan(x e^x) \]

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