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Trigonometric integrals

Integral of cot x: Formula, Derivation and Examples

The integral of cot x is ln|sin x| + C. See the u-substitution with u = sin x, how it pairs with tan x, plus cot(ax), cot² x and definite integral examples.

Trigonometric integrals Updated 2 min read

At a glance

\int \cot x\,dx = \ln\lvert\sin x\rvert + C
Antiderivative
\ln\lvert\sin x\rvert + C
Equivalent form
-\ln\lvert\csc x\rvert + C
Method
Rewrite as \dfrac{\cos x}{\sin x}, then u = \sin x
Linear form
\int\cot(ax)\,dx = \dfrac{1}{a}\ln\lvert\sin(ax)\rvert + C
Check
\dfrac{d}{dx}\ln\lvert\sin x\rvert = \cot x
On this page
  1. Graph of cot x and its antiderivative
  2. Derivation, step by step
  3. Check by differentiating
  4. Comparing tan and cot
  5. Definite integrals
  6. Substitution variants
  7. Common mistakes

The integral of \cot x is \ln\lvert\sin x\rvert + C. It comes from one substitution and is the mirror image of the integral of tan x.

\int \cot x\,dx = \ln\lvert\sin x\rvert + C

Graph of cot x and its antiderivative

Area under y = cot x from π/6 to π/2, with the antiderivative y = ln|sin x|
The shaded area is \ln 2 \approx 0.693. The dashed antiderivative reaches its maximum of 0 at x = \frac{\pi}{2}, exactly where \cot x crosses zero.

Derivation, step by step

  1. Rewrite: \displaystyle\int\cot x\,dx = \int\frac{\cos x}{\sin x}\,dx.
  2. Let u = \sin x. Then du = \cos x\,dx.
  3. Substitute: \displaystyle\int\frac{du}{u} = \ln\lvert u\rvert + C.
  4. Put u = \sin x back: \ln\lvert\sin x\rvert + C.

Step 3 is the integral of 1/x. Because \ln\lvert\sin x\rvert = -\ln\lvert\csc x\rvert, some books write the answer as -\ln\lvert\csc x\rvert + C. It is the same function.

Check by differentiating

\frac{d}{dx}\ln\lvert\sin x\rvert = \frac{\cos x}{\sin x} = \cot x \checkmark

The forward direction is different: the derivative of cot x is -\csc^2 x.

Comparing tan and cot

\tan x \cot x
Rewrite \frac{\sin x}{\cos x} \frac{\cos x}{\sin x}
Substitution u = \cos x, du = -\sin x\,dx u = \sin x, du = \cos x\,dx
Integral -\ln\lvert\cos x\rvert + C \ln\lvert\sin x\rvert + C
Derivative \sec^2 x -\csc^2 x

The minus sign shows up for \tan x but not for \cot x, because \cos x differentiates to -\sin x while \sin x differentiates to +\cos x.

Definite integrals

\int_{\pi/6}^{\pi/2}\cot x\,dx = \ln 1 - \ln\frac{1}{2} = \ln 2
\int_{\pi/4}^{\pi/2}\cot x\,dx = 0 - \ln\frac{\sqrt{2}}{2} = \frac{1}{2}\ln 2

Starting at x = 0 gives a divergent improper integral, since \ln\sin x \to -\infty as x \to 0^+.

Substitution variants

\int\cot(ax + b)\,dx = \frac{1}{a}\ln\lvert\sin(ax + b)\rvert + C

Example 1. \int\cot(4x)\,dx = \frac{1}{4}\ln\lvert\sin 4x\rvert + C.

Example 2. \int x\cot(x^2)\,dx = \frac{1}{2}\ln\lvert\sin(x^2)\rvert + C, with u = x^2.

Example 3: cot squared. Use \cot^2 x = \csc^2 x - 1:

\int\cot^2 x\,dx = \int(\csc^2 x - 1)\,dx = -\cot x - x + C

Check: \frac{d}{dx}(-\cot x - x) = \csc^2 x - 1 = \cot^2 x.

Example 4: a hidden cot. \int\frac{\cos x}{3\sin x + 2}\,dx = \frac{1}{3}\ln\lvert 3\sin x + 2\rvert + C, with u = 3\sin x + 2.

Common mistakes

  1. Answering -\csc^2 x. That is the derivative of \cot x.
  2. Adding a minus sign by analogy with \tan x. -\ln\lvert\sin x\rvert has derivative -\cot x.
  3. Answering \ln\lvert\cos x\rvert. The log goes on the denominator of \frac{\cos x}{\sin x}, which is \sin x.
  4. Dropping the absolute value. \sin x < 0 on (\pi, 2\pi).

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Frequently asked questions

What is the integral of cot x?
The integral of \cot x is \ln\lvert\sin x\rvert + C.

What is the integral of cos x / sin x?
That is \cot x, so the integral is \ln\lvert\sin x\rvert + C.

What is the integral of cot(2x)?
It is \frac{1}{2}\ln\lvert\sin 2x\rvert + C.

What is the integral of cot x from π/6 to π/2?
It is \ln\sin\frac{\pi}{2} - \ln\sin\frac{\pi}{6} = 0 - \ln\frac{1}{2} = \ln 2 \approx 0.693.

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Related integrals

Integral of tan x \int \tan x\,dx = \ln\lvert\sec x\rvert + C The integral of tan x is ln|sec x| + C, or -ln|cos x| + C. See the u-substitution with u = cos x, why both forms agree, and worked definite integrals. Integral of csc x \int \csc x\,dx = -\ln\lvert\csc x + \cot x\rvert + C The integral of csc x is -ln|csc x + cot x| + C, also written ln|csc x - cot x| or ln|tan(x/2)|. See the derivation, why the forms agree and examples. Integral of sin x \int \sin x\,dx = -\cos x + C The integral of sin x is -cos x + C. See where the minus sign comes from, the area of one arch, sin(ax + b) by substitution and worked definite integrals.