Integrate the following :
(ii) `int(1)/(1-cos x)dx`

Answer

To solve the integral \( \int \frac{1}{1 - \cos x} \, dx \), we can follow these steps:

### Step 1: Rewrite the integrand using trigonometric identities
We know from trigonometric identities that:
\[ 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \]
Thus, we can rewrite the integral as:
\[ \int \frac{1}{1 - \cos x} \, dx = \int \frac{1}{2 \sin^2\left(\frac{x}{2}\right)} \, dx \]

### Step 2: Simplify the integral
This can be simplified to:
\[ \int \frac{1}{2 \sin^2\left(\frac{x}{2}\right)} \, dx = \frac{1}{2} \int \csc^2\left(\frac{x}{2}\right) \, dx \]

### Step 3: Use substitution
Let \( u = \frac{x}{2} \). Then, \( dx = 2 \, du \). Substituting this into the integral gives:
\[ \frac{1}{2} \int \csc^2(u) \cdot 2 \, du = \int \csc^2(u) \, du \]

### Step 4: Integrate
The integral of \( \csc^2(u) \) is:
\[ \int \csc^2(u) \, du = -\cot(u) + C \]
Substituting back \( u = \frac{x}{2} \):
\[ -\cot\left(\frac{x}{2}\right) + C \]

### Step 5: Final result
Thus, the final result of the integral is:
\[ \int \frac{1}{1 - \cos x} \, dx = -\cot\left(\frac{x}{2}\right) + C \]

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