`int x. a^(x) dx `

Answer

To evaluate the integral \( \int x a^x \, dx \), we will use the integration by parts formula. The integration by parts formula states that:

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

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

Let:
- \( u = x \) (First function)
- \( dv = a^x \, dx \) (Second function)

### Step 2: Differentiate \( u \) and integrate \( dv \)

Now we differentiate \( u \) and integrate \( dv \):
- \( du = dx \)
- To find \( v \), we 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 found:
\[ \int x a^x \, dx = x \cdot \frac{a^x}{\ln a} - \int \frac{a^x}{\ln a} \, dx \]

### Step 4: Simplify the integral

Now we need to evaluate 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 we substitute this back into our equation:
\[ \int x a^x \, dx = x \cdot \frac{a^x}{\ln a} - \frac{1}{\ln a} \cdot \frac{a^x}{(\ln a)^2} \]

### Step 6: Combine the terms

Combining the terms gives us:
\[ \int x a^x \, dx = \frac{x a^x}{\ln a} - \frac{a^x}{(\ln a)^2} + C \]
where \( C \) is the constant of integration.

### Final Answer

Thus, the final result is:
\[ \int x a^x \, dx = \frac{a^x}{\ln a} \left( x - \frac{1}{\ln a} \right) + C \]