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Integral of cos⁡(x)\cos\left(x\right)

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

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Graph of f(x) = cos(x) with the area from 0 to π/2 shaded-0.50.511.522.5-0.50.51f(x)
Graph of f(x) = cos(x) with the area from 0 to π/2 shaded. The shaded area equals the definite integral, about 1.

How to work it out

  1. 1

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

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

    Method: standard integral.

  3. 3

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

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

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

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

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and π/2:

    ∫0π2cos⁡(x) dx=[sin⁡(x)]0π2≈1\int_{0}^{\frac{\pi}{2}} \cos\left(x\right)\,dx = \Big[\sin\left(x\right)\Big]_{0}^{\frac{\pi}{2}} \approx 1
LaTeX source of the formulas on this page
\[ \int \cos\left(x\right)\,dx = \sin\left(x\right) + C \]
\[ \int \cos\left(x\right)\,dx \]
\[ \frac{d}{dx}\left[\sin\left(x\right)\right] = \cos\left(x\right) \]
\[ \int_{0}^{\frac{\pi}{2}} \cos\left(x\right)\,dx = \Big[\sin\left(x\right)\Big]_{0}^{\frac{\pi}{2}} \approx 1 \]

Definite integrals of the same function

IntervalValue
from 0 to π/21
from 0 to 0.78539816339744830.70710678

Questions

What is the integral of cos(x)?

The integral of cos⁡(x)\cos\left(x\right) is sin⁡(x)+C\sin\left(x\right) + 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(x) from 0 to π/2?

About 1.

More integrals

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