Evaluate the integral: integral cot² x dx
Evaluate the integral: $$ \int \cot^2 x \, dx. $$
Explanation of Method
Just as $\tan^2 x = \sec^2 x - 1$ handles powers of tangent, the companion Pythagorean identity $\cot^2 x = \csc^2 x - 1$ handles powers of cotangent. This turns the integral into a difference of the standard derivative-of-cotangent form $\csc^2 x$ and the constant function $1$.
- Rewrite $\cot^2 x$ using the Pythagorean identity.
- Split the integral into two simpler integrals.
- Integrate each term using standard results.
- Combine the results and add the constant of integration.
Step 1 · Use the trigonometric identity for $\cot^2 x$
The trigonometric identity for $\cot^2 x$ is:
$$ \cot^2 x = \csc^2 x - 1. $$
Substituting this identity into the integral, we have:
$$ \int \cot^2 x\,dx = \int (\csc^2 x - 1)\,dx. $$
Step 2 · Split the integral
The integral is now split into two simpler integrals:
$$ \int \cot^2 x\,dx = \int \csc^2 x\,dx - \int 1\,dx. $$
First term: $\displaystyle \int \csc^2 x\,dx$ is a standard result:
$$ \int \csc^2 x\,dx = -\cot x. $$
Second term: $\displaystyle \int 1\,dx$:
$$ \int 1\,dx = x. $$
Step 3 · Combine the results
Combining the results of the two integrals:
$$ \int \cot^2 x\,dx = -\cot x - x + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Antiderivative |
|---|---|
| $\csc^2 x$ | $-\cot x$ |
| $-1$ | $-x$ |
Additional Notes
- The domain of $I$ excludes $x = k\pi$ for integer $k$, where $\cot x$ and $\csc^2 x$ are undefined.
- Compare with Problem 18: the identity $\cot^2 x = \csc^2 x-1$ mirrors $\tan^2 x=\sec^2 x-1$, but the antiderivative of $\csc^2 x$ carries a negative sign, $-\cot x$, unlike $\int \sec^2 x\,dx=\tan x$.
- The result can be checked by differentiating $-\cot x - x$ and confirming it returns $\csc^2 x - 1 = \cot^2 x$.
Result
The solution to the integral is: