`int(e^(x)(1+x))/(cos^(2)(e^(x)x))dx` equal to

Options

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

Explanation

To solve the integral

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

we can follow these steps:

### Step 1: Rewrite the Integral
We can rewrite the integral as:

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

### Step 2: Substitution
Let’s make a substitution. Let

\[ t = e^x x. \]

Now, we need to find \( dt \). We differentiate \( t \) with respect to \( x \):

Using the product rule,

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

Thus, we have:

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

### Step 3: Substitute in the Integral
Now we substitute \( t \) and \( dx \) into the integral:

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

The \( e^x (1 + x) \) terms cancel out:

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

### Step 4: Simplify the Integral
The integral of \( \frac{1}{\cos^2(t)} \) is:

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

### Step 5: Integrate
The integral of \( \sec^2(t) \) is:

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

### Step 6: Substitute Back
Now we substitute back \( t = e^x x \):

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

### Final Result
Thus, the integral

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

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