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Integral of cos⁡(2x)\cos\left(2 x\right)

The integral of cos⁡(2x)\cos\left(2 x\right) is sin⁡(2x)2+C\frac{\sin\left(2 x\right)}{2} + C.

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Graph of f(x) = cos(2x) with the area from 0 to π shaded-112345-1-0.50.51f(x)
Graph of f(x) = cos(2x) with the area from 0 to π shaded. The shaded area equals the definite integral, about -1.2246468e-16.

How to work it out

  1. 1

    We want an antiderivative of f(x) = cos(2x).

    ∫cos⁡(2x) dx\int \cos\left(2 x\right)\,dx
  2. 2

    Method: ∫cos(kx) dx = sin(kx)/k.

  3. 3

    That gives the antiderivative; add the constant C because any constant disappears when you differentiate:

    ∫cos⁡(2x) dx=sin⁡(2x)2+C\int \cos\left(2 x\right)\,dx = \frac{\sin\left(2 x\right)}{2} + C
  4. 4

    Check by differentiating the answer; you must get the original function back:

    ddx[sin⁡(2x)2]=4cos⁡(2x)4\frac{d}{dx}\left[\frac{\sin\left(2 x\right)}{2}\right] = \frac{4 \cos\left(2 x\right)}{4}
  5. 5

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and π:

    ∫0πcos⁡(2x) dx=[sin⁡(2x)2]0π≈−1.2246468e−16\int_{0}^{\pi} \cos\left(2 x\right)\,dx = \Big[\frac{\sin\left(2 x\right)}{2}\Big]_{0}^{\pi} \approx -1.2246468e-16
LaTeX source of the formulas on this page
\[ \int \cos\left(2 x\right)\,dx = \frac{\sin\left(2 x\right)}{2} + C \]
\[ \int \cos\left(2 x\right)\,dx \]
\[ \frac{d}{dx}\left[\frac{\sin\left(2 x\right)}{2}\right] = \frac{4 \cos\left(2 x\right)}{4} \]
\[ \int_{0}^{\pi} \cos\left(2 x\right)\,dx = \Big[\frac{\sin\left(2 x\right)}{2}\Big]_{0}^{\pi} \approx -1.2246468e-16 \]

Definite integrals of the same function

IntervalValue
from 0 to π-1.2246468e-16
from 0 to π/26.123234e-17

Questions

What is the integral of cos(2x)?

The integral of cos⁡(2x)\cos\left(2 x\right) is sin⁡(2x)2+C\frac{\sin\left(2 x\right)}{2} + C.

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of cos(2x) from 0 to π?

About -1.2246468e-16.

More integrals

    Integral of cos(2x): sin(2x) / 2 + C (with steps) | WordGPT