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Integral of sin x

∫sin⁡(x) dx=−cos⁡(x)+C\int \sin\left(x\right)\,dx = -\cos\left(x\right) + C

Since the derivative of cos x is −sin x, the antiderivative of sin x is −cos x. Over one full period the signed area is zero.

Step by step

  1. Trig rule

    ∫sin u du = −cos u.

    ∫sin⁡(x) dx=−cos⁡(x)\int \sin\left(x\right)\,dx = -\cos\left(x\right)
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  • f(x) = sin x
  • F(x) + C

Every choice of C gives a valid antiderivative, since shifting up or down doesn’t change the slope.

Open in the integral calculator

Questions students ask

What is the integral of sin x?

∫sin⁡(x) dx=−cos⁡(x)+C\int \sin\left(x\right)\,dx = -\cos\left(x\right) + C.

How do you integrate sin x?

Since the derivative of cos x is −sin x, the antiderivative of sin x is −cos x. Over one full period the signed area is zero.

How can I check the answer?

Differentiate it: the derivative of −cos⁡(x)-\cos\left(x\right) is sin⁡(x)\sin\left(x\right). CalcViz checks every antiderivative this way before showing it.