(i) `int(6x^(5) - 2/(x^(4)) - 7x + 3/x - 5 + 4e^(x) + 7^(x))dx`
(ii) `int (8-x+2x^(3) -6/(x^(3)) +2x^(-5) + 5x^(-1)) dx` , (iii) `int (x/a+a/x+x^(a) +a^(x) + ax) dx`

Answer

Let's solve the three integration problems step by step.

### (i) \(\int (6x^{5} - \frac{2}{x^{4}} - 7x + \frac{3}{x} - 5 + 4e^{x} + 7^{x}) \, dx\)

1. **Integrate each term separately:**
- \(\int 6x^{5} \, dx = 6 \cdot \frac{x^{6}}{6} = x^{6}\)
- \(\int -\frac{2}{x^{4}} \, dx = -2 \cdot \int x^{-4} \, dx = -2 \cdot \frac{x^{-3}}{-3} = \frac{2}{3x^{3}}\)
- \(\int -7x \, dx = -7 \cdot \frac{x^{2}}{2} = -\frac{7}{2}x^{2}\)
- \(\int \frac{3}{x} \, dx = 3 \ln |x|\)
- \(\int -5 \, dx = -5x\)
- \(\int 4e^{x} \, dx = 4e^{x}\)
- \(\int 7^{x} \, dx = \frac{7^{x}}{\ln(7)}\)

2. **Combine all the results:**
\[ \int (6x^{5} - \frac{2}{x^{4}} - 7x + \frac{3}{x} - 5 + 4e^{x} + 7^{x}) \, dx = x^{6} + \frac{2}{3x^{3}} - \frac{7}{2}x^{2} + 3 \ln |x| - 5x + 4e^{x} + \frac{7^{x}}{\ln(7)} + C \]

### (ii) \(\int (8 - x + 2x^{3} - \frac{6}{x^{3}} + 2x^{-5} + 5x^{-1}) \, dx\)

1. **Integrate each term separately:**
- \(\int 8 \, dx = 8x\)
- \(\int -x \, dx = -\frac{x^{2}}{2}\)
- \(\int 2x^{3} \, dx = 2 \cdot \frac{x^{4}}{4} = \frac{1}{2}x^{4}\)
- \(\int -\frac{6}{x^{3}} \, dx = -6 \cdot \int x^{-3} \, dx = -6 \cdot \frac{x^{-2}}{-2} = 3x^{-2}\)
- \(\int 2x^{-5} \, dx = 2 \cdot \frac{x^{-4}}{-4} = -\frac{1}{2}x^{-4}\)
- \(\int 5x^{-1} \, dx = 5 \ln |x|\)

2. **Combine all the results:**
\[ \int (8 - x + 2x^{3} - \frac{6}{x^{3}} + 2x^{-5} + 5x^{-1}) \, dx = 8x - \frac{x^{2}}{2} + \frac{1}{2}x^{4} + 3x^{-2} - \frac{1}{2}x^{-4} + 5 \ln |x| + C \]

### (iii) \(\int \left(\frac{x}{a} + \frac{a}{x} + x^{a} + a^{x} + ax\right) \, dx\)

1. **Integrate each term separately:**
- \(\int \frac{x}{a} \, dx = \frac{1}{a} \cdot \frac{x^{2}}{2} = \frac{x^{2}}{2a}\)
- \(\int \frac{a}{x} \, dx = a \ln |x|\)
- \(\int x^{a} \, dx = \frac{x^{a+1}}{a+1}\)
- \(\int a^{x} \, dx = \frac{a^{x}}{\ln(a)}\)
- \(\int ax \, dx = a \cdot \frac{x^{2}}{2} = \frac{ax^{2}}{2}\)

2. **Combine all the results:**
\[ \int \left(\frac{x}{a} + \frac{a}{x} + x^{a} + a^{x} + ax\right) \, dx = \frac{x^{2}}{2a} + a \ln |x| + \frac{x^{a+1}}{a+1} + \frac{a^{x}}{\ln(a)} + \frac{ax^{2}}{2} + C \]