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

Options

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

Answer

To solve the integral

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

we will use substitution and integration techniques. Here’s the step-by-step solution:

### Step 1: Substitution
Let \( t = x e^x \).

To differentiate \( t \), we apply the product rule:

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

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

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

### Step 2: Rewrite the Integral
Now, substituting \( t \) into the integral, we have:

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

The \( e^x(1+x) \) in the numerator and denominator cancels out:

\[ \int \frac{1}{\cos^2(t)} \, dt. \]

### Step 3: Recognize the Integral
The integral \( \frac{1}{\cos^2(t)} \) is known to be:

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

### Step 4: Back Substitute
Now, we substitute back \( t = x e^x \):

\[ \tan(t) = \tan(x e^x). \]

Thus, the final answer is:

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

### Final Solution
The solution to the integral is:

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

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