Evaluate: `inta^(m x)b^(n x)dx`

Explanation

To evaluate the integral

\[ \int a^{mx} b^{nx} \, dx \]

we can follow these steps:

### Step 1: Rewrite the integrand
We can rewrite the integrand \( a^{mx} b^{nx} \) as follows:

\[ a^{mx} b^{nx} = (a^m)^{x} (b^n)^{x} = (a^m b^n)^{x} \]

### Step 2: Use substitution
Let \( k = a^m b^n \). Then the integral becomes:

\[ \int (k)^{x} \, dx \]

### Step 3: Integrate
The integral of \( k^x \) is given by:

\[ \int k^x \, dx = \frac{k^x}{\ln(k)} + C \]

### Step 4: Substitute back
Now, substituting back for \( k \):

\[ \int a^{mx} b^{nx} \, dx = \frac{(a^m b^n)^{x}}{\ln(a^m b^n)} + C \]

### Step 5: Simplify
We can simplify the logarithm in the denominator:

\[ \ln(a^m b^n) = m \ln(a) + n \ln(b) \]

Thus, the final answer is:

\[ \int a^{mx} b^{nx} \, dx = \frac{(a^m b^n)^{x}}{m \ln(a) + n \ln(b)} + C \]

### Final Answer
\[ \int a^{mx} b^{nx} \, dx = \frac{(a^m b^n)^{x}}{m \ln(a) + n \ln(b)} + C \]

---