Integrate the following functions
(i) `int((1+x)^(4))/(x)dx`
(ii) `int(e^(200x)+e^(202x))/(e^(x)+e^(-x))dx`
(iii) `int(1+tan^(2)x)/(1+cot^(2)x)dx`

Answer

Let's solve the given integrals step by step.

### (i) Integrate \( \int \frac{(1+x)^4}{x} \, dx \)

1. **Rewrite the integrand**:
\[ \frac{(1+x)^4}{x} = \frac{(1+x)^4}{x} = \frac{(1+x)^4}{x} = (1+x)^4 \cdot \frac{1}{x} \]

2. **Expand \( (1+x)^4 \)** using the binomial theorem:
\[ (1+x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 \]

3. **Substitute back into the integral**:
\[ \int \frac{(1+x)^4}{x} \, dx = \int \left(\frac{1}{x} + 4 + 6x + 4x^2 + x^3\right) \, dx \]

4. **Integrate term by term**:
\[ = \int \frac{1}{x} \, dx + \int 4 \, dx + \int 6x \, dx + \int 4x^2 \, dx + \int x^3 \, dx \]
\[ = \ln |x| + 4x + 3x^2 + \frac{4}{3}x^3 + \frac{1}{4}x^4 + C \]

### Final answer for (i):
\[ \int \frac{(1+x)^4}{x} \, dx = \ln |x| + 4x + 3x^2 + \frac{4}{3}x^3 + \frac{1}{4}x^4 + C \]