Integral of cos x: Formula, Derivation and Examples

The integral of \cos x is \sin x + C. Because the derivative of \sin x is exactly \cos x, this is one of the cleanest integrals there is.

Graph of cos x and its antiderivative

The antiderivative \sin x peaks at x = \frac{\pi}{2}, exactly where \cos x crosses zero from positive to negative.

Derivation, step by step

1. Ask: which function has derivative \cos x?
2. The derivative of sin x is \cos x.
3. So \sin x is an antiderivative of \cos x.
4. Add C to include every antiderivative: \sin x + C.

No sign fix is needed here, unlike the integral of sin x.

Check by differentiating

Definite integrals

Since \cos x is even, \int_{-a}^{a}\cos x\,dx = 2\int_0^a\cos x\,dx = 2\sin a.

Substitution variants

With u = ax + b:

For example, \int\cos(4x - 1)\,dx = \frac{1}{4}\sin(4x - 1) + C, and \int\cos(\pi x)\,dx = \frac{1}{\pi}\sin(\pi x) + C.

When the inside derivative appears as a factor, use u-substitution:

For the last one, du = \frac{1}{2\sqrt{x}}dx, so \frac{dx}{\sqrt{x}} = 2\,du.

Worked examples

Example 1. \int(2\cos x - 3\sin x)\,dx = 2\sin x + 3\cos x + C. Note how the minus in front of \sin combines with the minus in -\cos x to give +3\cos x.

Example 2. \int_0^{\pi/4}\cos(2x)\,dx = \left[\frac{1}{2}\sin(2x)\right]_0^{\pi/4} = \frac{1}{2}\sin\frac{\pi}{2} = \frac{1}{2}.

Example 3: an initial value problem. If f'(x) = \cos x and f(0) = 3, then f(x) = \sin x + C and f(0) = C = 3, so f(x) = \sin x + 3.

Example 4: simple harmonic motion. If a spring has velocity v(t) = 4\cos(2t) and position s(0) = 0, then s(t) = 2\sin(2t). The amplitude is 2.

Common mistakes

1. Writing -\sin x. That is the derivative of \cos x, not the integral.
2. Forgetting to divide by a. \int\cos(3x)\,dx = \frac{1}{3}\sin(3x) + C, not \sin(3x) + C.
3. Integrating \cos^2 x as \sin^2 x or \frac{\cos^3 x}{3}. Powers of trig functions need identities. See the integral of cos² x.
4. Integrating \cos(x^2) as \frac{\sin(x^2)}{2x}. You cannot divide by a variable inside derivative. \int\cos(x^2)\,dx has no elementary antiderivative.

Practise with trig integral problems.