`int((x+1)e^(x))/(cos^(2)(xe^(x)))dx=?`

Options

A. `tan(xe^(x))+C`
B. `-tan(xe^(x))+C`
C. `cot(xe^(x))+C`
D. none of these

Explanation

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

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

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

From this, we can express \( dx \) in terms of \( dt \):

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

### Step 2: Substitute in the Integral
Now, we substitute \( t \) and \( dx \) into the integral:

\[ \int \frac{(x+1)e^x}{\cos^2(x e^x)} \, dx = \int \frac{(x+1)e^x}{\cos^2(t)} \cdot \frac{dt}{e^x(1+x)} \]

Notice that the \( e^x \) and \( (x+1) \) terms cancel out:

\[ = \int \frac{dt}{\cos^2(t)} \]

### Step 3: Simplify the Integral
The integral \(\int \frac{dt}{\cos^2(t)}\) can be rewritten using the identity \(\frac{1}{\cos^2(t)} = \sec^2(t)\):

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

### Step 4: Integrate
The integral of \(\sec^2(t)\) is:

\[ = \tan(t) + C \]

### Step 5: Substitute Back
Now, we substitute back \( t = x e^x \):

\[ = \tan(x e^x) + C \]

### Final Answer
Thus, the final answer is:

\[ \int \frac{(x+1)e^x}{\cos^2(x e^x)} \, dx = \tan(x e^x) + C \]

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