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Trigonometric integrals

Integral of cos² x: Formula, Derivation and Examples

The integral of cos² x is x/2 + sin(2x)/4 + C. See the power-reducing identity, a check, cos²(ax) and the area of a circle.

Trigonometric integrals Updated 2 min read

At a glance

\int \cos^2 x\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C
Antiderivative
\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C
Identity used
\cos^2 x = \dfrac{1 + \cos 2x}{2}
Equivalent form
\dfrac{x + \sin x\cos x}{2} + C
From 0 to 2π
\pi
Check with sin²
\int(\sin^2 x + \cos^2 x)\,dx = x + C
On this page
  1. Graph of cos² x and its antiderivative
  2. Derivation, step by step
  3. Check by differentiating
  4. Definite integrals
  5. Substitution variants
  6. Where it shows up: the area of a circle
  7. Worked example: cos² x sin² x
  8. Common mistakes

The integral of \cos^2 x is \dfrac{x}{2} + \dfrac{\sin 2x}{4} + C. Like \sin^2 x, it needs a power-reducing identity before you integrate.

\int \cos^2 x\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C

Graph of cos² x and its antiderivative

Area under y = cos² x from -π/2 to π/2, with the antiderivative y = x/2 + sin(2x)/4
The shaded area is \frac{\pi}{2} \approx 1.571. The dashed antiderivative rises fastest at x = 0, where \cos^2 x = 1, and flattens where \cos^2 x = 0.

Derivation, step by step

  1. Start from \cos 2x = 2\cos^2 x - 1.
  2. Solve for \cos^2 x: \cos^2 x = \dfrac{1 + \cos 2x}{2}.
  3. Substitute: \displaystyle\int\cos^2 x\,dx = \frac{1}{2}\int(1 + \cos 2x)\,dx.
  4. Integrate: \frac{1}{2}\left(x + \frac{\sin 2x}{2}\right) + C.
  5. Simplify: \frac{x}{2} + \frac{\sin 2x}{4} + C.

Step 4 uses the integral of cos x with a factor of \frac{1}{2} from the inside 2x.

Check by differentiating

\frac{d}{dx}\left(\frac{x}{2} + \frac{\sin 2x}{4}\right) = \frac{1}{2} + \frac{\cos 2x}{2} = \cos^2 x \checkmark

A second check: add this answer to the integral of sin² x:

\left(\frac{x}{2} + \frac{\sin 2x}{4}\right) + \left(\frac{x}{2} - \frac{\sin 2x}{4}\right) = x

That matches \int(\sin^2 x + \cos^2 x)\,dx = \int 1\,dx = x. The forward direction is on the derivative of cos² x page.

Definite integrals

\int_0^{\pi/2}\cos^2 x\,dx = \frac{\pi}{4}, \qquad \int_0^{\pi}\cos^2 x\,dx = \frac{\pi}{2}, \qquad \int_0^{2\pi}\cos^2 x\,dx = \pi
\int_0^{\pi/6}\cos^2 x\,dx = \frac{\pi}{12} + \frac{\sin(\pi/3)}{4} = \frac{\pi}{12} + \frac{\sqrt{3}}{8} \approx 0.478

Substitution variants

\int\cos^2(ax)\,dx = \frac{x}{2} + \frac{\sin(2ax)}{4a} + C

Example 1. \int\cos^2(5x)\,dx = \frac{x}{2} + \frac{\sin 10x}{20} + C.

Example 2. \int\cos^2\frac{x}{2}\,dx = \int\frac{1 + \cos x}{2}\,dx = \frac{x}{2} + \frac{\sin x}{2} + C.

Example 3: with a sin x factor. \int\cos^2 x\sin x\,dx = -\frac{\cos^3 x}{3} + C, using u = \cos x. No identity needed.

Where it shows up: the area of a circle

The area of a circle of radius 1 is 4\int_0^1\sqrt{1 - x^2}\,dx. Substitute x = \sin\theta, dx = \cos\theta\,d\theta:

4\int_0^{\pi/2}\cos\theta\cdot\cos\theta\,d\theta = 4\int_0^{\pi/2}\cos^2\theta\,d\theta = 4\cdot\frac{\pi}{4} = \pi

The same steps give the integral of √(1 - x²). See trig substitution for the method.

Worked example: cos² x sin² x

Use \sin x\cos x = \frac{\sin 2x}{2}, so \sin^2 x\cos^2 x = \frac{\sin^2 2x}{4} = \frac{1 - \cos 4x}{8}. Then

\int\sin^2 x\cos^2 x\,dx = \frac{x}{8} - \frac{\sin 4x}{32} + C

Common mistakes

  1. Answering \sin^2 x. Its derivative is 2\sin x\cos x.
  2. Answering \frac{\cos^3 x}{3}. That would need a -\sin x factor.
  3. Using 1 - \cos 2x. That identity is for \sin^2 x. A quick test: at x = 0, \cos^2 x = 1, and \frac{1 + \cos 0}{2} = 1.
  4. Writing \frac{\sin 2x}{2} instead of \frac{\sin 2x}{4}. Remember both halvings: one from the identity, one from the inside 2x.

Practise on trig integral problems.

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Frequently asked questions

What is the integral of cos² x?
It is \frac{x}{2} + \frac{\sin 2x}{4} + C, or equivalently \frac{1}{2}(x + \sin x\cos x) + C.

What is the integral of cos² x from 0 to 2π?
It is \pi. The \sin 2x term is 0 at both ends, leaving \frac{2\pi}{2}.

What is the integral of cos²(x/2)?
Since \cos^2\frac{x}{2} = \frac{1 + \cos x}{2}, the integral is \frac{x}{2} + \frac{\sin x}{2} + C.

What is the integral of cos² x sin x?
Use u = \cos x, du = -\sin x\,dx. The answer is -\frac{\cos^3 x}{3} + C.

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Related integrals

Integral of sin² x \int \sin^2 x\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C The integral of sin² x is x/2 - sin(2x)/4 + C. See the power-reducing identity step by step, a check by differentiating, sin²(ax) and definite integrals. Integral of cos x \int \cos x\,dx = \sin x + C The integral of cos x is sin x + C. See why there is no minus sign, the area of one hump, cos(ax + b) by substitution and worked definite integral examples. Integral of √(1−x²) \int \sqrt{1-x^2}\,dx = \frac{1}{2}\left(x\sqrt{1-x^2} + \arcsin x\right) + C The integral of √(1−x²) is ½(x√(1−x²) + arcsin x) + C. Learn the trig substitution, the link to the area of a circle and worked examples.