What is the process for integrating cot^2(x)dx?

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To integrate the function cot2(x)dx, we can use a well-known trigonometric identity and substitution method to simplify our work. Here’s a detailed step-by-step explanation:

  1. Recall the identity for cot2(x): cot2(x) = csc2(x) - 1. This is useful because it allows us to express cot2(x) in terms of csc2(x).
  2. Using this identity, we rewrite the integral: ∫ cot2(x) dx = ∫ (csc2(x) - 1) dx.
  3. Now we can separate the integral into two parts: ∫ cot2(x) dx = ∫ csc2(x) dx - ∫ dx.
  4. Next, we find the integral of csc2(x). The integral of csc2(x) is a standard result: ∫ csc2(x) dx = -cot(x) + C, where C is the constant of integration.
  5. The integral of 1 is simply x, so: ∫ dx = x.
  6. Combining these results, we’ve got:

    ∫ cot2(x) dx = -cot(x) - x + C.

In summary, the result of integrating cot2(x)dx is:

∫ cot2(x) dx = -cot(x) - x + C

Now you have a neat and clear way to integrate cot2(x)dx! Whether for studying calculus or tackling some mathematical tasks, this foundational technique can be quite beneficial.

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