`int e^x sin(e^x) dx=`

Options

A. `cos(e^x)+c`
B. `cos(e^x)+c`
C. `sin(e^x)+c`
D. `-sin (e^x)+c`

Answer

To solve the integral \( \int e^x \sin(e^x) \, dx \), we can use the substitution method. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = e^x \). Then, the differential \( dt \) can be expressed as:
\[ dt = e^x \, dx \quad \Rightarrow \quad dx = \frac{dt}{e^x} = \frac{dt}{t} \]

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

### Step 3: Integrate
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) + C \]

### Step 4: Substitute Back
Now we substitute back \( t = e^x \):
\[ -\cos(t) + C = -\cos(e^x) + C \]

### Final Answer
Thus, the final answer is:
\[ \int e^x \sin(e^x) \, dx = -\cos(e^x) + C \]