How do you integrate x – sin(x) by 1 – cos(x)?

by

To integrate the expression x – sin(x) divided by 1 – cos(x), we start with the integral:

$$\int \frac{x – \sin{x}}{1 – \cos{x}} dx$$

This integral can be tackled through substitution or integration techniques. Let’s break it down:

  1. First, recognize the integral splits into two parts:
    • $$\int \frac{x}{1 – \cos{x}} dx$$
    • $$\int \frac{-\sin{x}}{1 – \cos{x}} dx$$
  2. For the first part, use integration by parts. Set:
    • u = x → du = dx
    • dv = \frac{1}{1 – \cos{x}} dx → v = \ln |\tan(\frac{x}{2})| + C (using the identity for integrating functions of this type).
  3. The formula for integration by parts is:

    4. $$\int u \, dv = uv – \int v \, du$$

  4. Now calculate uv and \int v \, du.
  5. For the second integral, $$\int \frac{-\sin{x}}{1 – \cos{x}} dx$$ can be transformed by observing that $$\frac{\sin{x}}{1 – \cos{x}} = \frac{1 – \cos{x}}{1 – \cos{x}} – \frac{\cos{x}}{1 – \cos{x}}$$. This allows us to simplify it further.

Combining both results will ultimately lead to a general solution. Always remember to simplify your results and express in terms of elementary functions if possible. You may also choose to use numerical methods for complex integrals.

Leave a Comment