Evaluate: `int(e^(3x)+e^(5x))/(e^x+e^(-x))dx`

Explanation

To solve the integral

\[ \int \frac{e^{3x} + e^{5x}}{e^x + e^{-x}} \, dx, \]

we can simplify the expression step by step.

### Step 1: Simplify the Denominator
The denominator can be rewritten using the property of exponents:

\[ e^x + e^{-x} = e^x + \frac{1}{e^x} = \frac{e^{2x} + 1}{e^x}. \]

### Step 2: Rewrite the Integral
Now, substituting this back into the integral gives:

\[ \int \frac{e^{3x} + e^{5x}}{\frac{e^{2x} + 1}{e^x}} \, dx = \int \frac{(e^{3x} + e^{5x}) e^x}{e^{2x} + 1} \, dx. \]

### Step 3: Simplify the Numerator
This simplifies to:

\[ \int \frac{e^{4x} + e^{6x}}{e^{2x} + 1} \, dx. \]

### Step 4: Factor Out Common Terms
We can factor out \(e^{4x}\) from the numerator:

\[ \int \frac{e^{4x}(1 + e^{2x})}{e^{2x} + 1} \, dx. \]

### Step 5: Cancel Common Terms
Now, we see that \(1 + e^{2x} = e^{2x} + 1\), thus we can cancel these terms:

\[ \int e^{4x} \, dx. \]

### Step 6: Integrate
Now we can integrate:

\[ \int e^{4x} \, dx = \frac{e^{4x}}{4} + C, \]

where \(C\) is the constant of integration.

### Final Answer
Thus, the final answer is:

\[ \frac{e^{4x}}{4} + C. \]