`int x. a^(x) dx `

Answer

To solve the integral \( \int x a^x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)

Let:
- \( u = x \) (which means \( du = dx \))
- \( dv = a^x \, dx \) (which means we need to find \( v \))

### Step 2: Find \( v \)

To find \( v \), we need to integrate \( dv \):

\[ v = \int a^x \, dx = \frac{a^x}{\ln a} \]

### Step 3: Apply the integration by parts formula

Now we can apply the integration by parts formula:

\[ \int x a^x \, dx = uv - \int v \, du \]

Substituting the values we have:

\[ \int x a^x \, dx = x \cdot \frac{a^x}{\ln a} - \int \frac{a^x}{\ln a} \, dx \]

### Step 4: Simplify the integral

Now, we need to compute the integral \( \int \frac{a^x}{\ln a} \, dx \):

\[ \int \frac{a^x}{\ln a} \, dx = \frac{1}{\ln a} \int a^x \, dx = \frac{1}{\ln a} \cdot \frac{a^x}{\ln a} = \frac{a^x}{(\ln a)^2} \]

### Step 5: Substitute back into the equation

Now substituting this back into our equation:

\[ \int x a^x \, dx = x \cdot \frac{a^x}{\ln a} - \frac{a^x}{(\ln a)^2} + C \]

### Step 6: Final result

Thus, the final result for the integral is:

\[ \int x a^x \, dx = \frac{a^x}{\ln a} \left( x - \frac{1}{\ln a} \right) + C \]