Write down the integral of the following (i) `1/x, x ne 0` (ii) `x+1/x , xne 0`

Answer

To solve the given problem, we need to find the integrals of the following functions:

(i) \( \frac{1}{x} \) where \( x \neq 0 \)

(ii) \( x + \frac{1}{x} \) where \( x \neq 0 \)

Let's solve them step by step.

### Step 1: Finding the integral of \( \frac{1}{x} \)

The integral we need to compute is:

\[ \int \frac{1}{x} \, dx \]

We know from calculus that the integral of \( \frac{1}{x} \) is the natural logarithm of the absolute value of \( x \):

\[ \int \frac{1}{x} \, dx = \log |x| + C \]

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

### Step 2: Finding the integral of \( x + \frac{1}{x} \)

Now, we need to compute:

\[ \int \left( x + \frac{1}{x} \right) \, dx \]

This can be separated into two integrals:

\[ \int x \, dx + \int \frac{1}{x} \, dx \]

Now, we compute each integral separately.

1. **Integral of \( x \)**:
\[ \int x \, dx = \frac{x^2}{2} + C_1 \]

2. **Integral of \( \frac{1}{x} \)** (already computed):
\[ \int \frac{1}{x} \, dx = \log |x| + C_2 \]

Now, combining both results:

\[ \int \left( x + \frac{1}{x} \right) \, dx = \frac{x^2}{2} + \log |x| + C \]

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

### Final Answers

(i) The integral of \( \frac{1}{x} \) is:

\[ \int \frac{1}{x} \, dx = \log |x| + C \]

(ii) The integral of \( x + \frac{1}{x} \) is:

\[ \int \left( x + \frac{1}{x} \right) \, dx = \frac{x^2}{2} + \log |x| + C \]