Evaluate :
`int (a^(x)+b^(x))/(c^(x))dx`

Answer

To evaluate the integral

\[ \int \frac{a^x + b^x}{c^x} \, dx, \]

we can start by rewriting the integrand.

### Step 1: Rewrite the integrand

We can express the integrand as:

\[ \frac{a^x + b^x}{c^x} = \frac{a^x}{c^x} + \frac{b^x}{c^x} = \left(\frac{a}{c}\right)^x + \left(\frac{b}{c}\right)^x. \]

### Step 2: Split the integral

Now, we can split the integral into two separate integrals:

\[ \int \left(\left(\frac{a}{c}\right)^x + \left(\frac{b}{c}\right)^x\right) \, dx = \int \left(\frac{a}{c}\right)^x \, dx + \int \left(\frac{b}{c}\right)^x \, dx. \]

### Step 3: Evaluate each integral

We know that the integral of \(k^x\) is given by:

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

Applying this formula to our integrals:

1. For the first integral, where \(k = \frac{a}{c}\):

\[ \int \left(\frac{a}{c}\right)^x \, dx = \frac{\left(\frac{a}{c}\right)^x}{\ln\left(\frac{a}{c}\right)} + C_1. \]

2. For the second integral, where \(k = \frac{b}{c}\):

\[ \int \left(\frac{b}{c}\right)^x \, dx = \frac{\left(\frac{b}{c}\right)^x}{\ln\left(\frac{b}{c}\right)} + C_2. \]

### Step 4: Combine the results

Now, we can combine the results of both integrals:

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

where \(C = C_1 + C_2\) is the constant of integration.

### Final Answer

Thus, the final answer is:

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