What is the value of `int(e^(x)(1+x))/(sin^(2)(xe^(x)))dx` ?

Options

A. A) `-e^(x)cotx+c`
B. B) `cos2(xe^(x))+c`
C. C) `logsin(xe^(x))+c`
D. D) `-cot(xe^(x))+c`

Answer

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

### Step 1: Substitution
Let \( t = x e^x \). To differentiate this, we apply the product rule:
\[ dt = e^x \, dx + x e^x \, dx = e^x(1 + x) \, dx \]
This implies:
\[ e^x(1 + x) \, dx = dt \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral in terms of \( t \):
\[ \int \frac{e^x(1+x)}{\sin^2(x e^x)} \, dx = \int \frac{1}{\sin^2(t)} \, dt \]

### Step 3: Simplify the Integral
The integral \( \int \frac{1}{\sin^2(t)} \, dt \) can be simplified using the identity:
\[ \frac{1}{\sin^2(t)} = \cot^2(t) \]
Thus, we have:
\[ \int \frac{1}{\sin^2(t)} \, dt = \int \cot^2(t) \, dt \]

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

### Step 5: Back Substitute
Now we substitute back \( t = x e^x \):
\[ -\cot(t) + C = -\cot(x e^x) + C \]

### Final Answer
Thus, the value of the integral is:
\[ -\cot(x e^x) + C \]

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