What is `int a^(x) e^(x) dx` equal to ?
where c is the constant of integration

Options

A. `(a^(x)e^(x))/(l na) + c`
B. `a^(x) e^(x) + c`
C. `(a^(x) e^(x))/(l n(ae)) + c`
D. None of the above

Explanation

To solve the integral \( \int a^x e^x \, dx \), we will use integration by parts. Let's go through the steps:

### Step 1: Set up the integral
Let:
\[ I = \int a^x e^x \, dx \]

### Step 2: Use integration by parts
We will use integration by parts, which states:
\[ \int u \, dv = uv - \int v \, du \]
Here, we can choose:
- \( u = a^x \) (which implies \( du = a^x \ln a \, dx \))
- \( dv = e^x \, dx \) (which implies \( v = e^x \))

### Step 3: Apply integration by parts
Now, applying the integration by parts formula:
\[ I = a^x e^x - \int e^x a^x \ln a \, dx \]
This simplifies to:
\[ I = a^x e^x - \ln a \int a^x e^x \, dx \]
Notice that the integral on the right side is the same as \( I \). Thus, we can rewrite it as:
\[ I = a^x e^x - \ln a \cdot I \]

### Step 4: Solve for \( I \)
Now, we can factor \( I \) out:
\[ I + \ln a \cdot I = a^x e^x \]
\[ I (1 + \ln a) = a^x e^x \]
Now, divide both sides by \( 1 + \ln a \):
\[ I = \frac{a^x e^x}{1 + \ln a} \]

### Step 5: Add the constant of integration
Finally, we include the constant of integration \( C \):
\[ \int a^x e^x \, dx = \frac{a^x e^x}{1 + \ln a} + C \]

### Final Answer
Thus, the integral \( \int a^x e^x \, dx \) is equal to:
\[ \frac{a^x e^x}{1 + \ln a} + C \]