MathematicsCalculusA-Level
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Integral of cos(x) Calculator

Antiderivative of the cosine function.

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Integral Value

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Overview

The integral of the cosine function represents the antiderivative that yields the sine function. In calculus, this operation determines the area under the cosine curve or the cumulative sum of its periodic values across a specified interval.

Symbols

Variables

I = Integral Value, x = Angle

Integral Value
Angle

Apply it well

When To Use

When to use: Use this integral when analyzing systems exhibiting simple harmonic motion, such as a vibrating string or a pendulum. It is essential when converting between acceleration, velocity, and position in physics for objects moving sinusoidally.

Why it matters: This relationship is a cornerstone of Fourier analysis, which decomposes complex signals into basic waves for telecommunications and audio processing. It also allows engineers to calculate power in AC circuits where voltage and current vary over time.

Avoid these traps

Common Mistakes

  • Adding negative sign.
  • Using degrees.

One free problem

Practice Problem

Find the value of the definite integral I = ∫ cos(t) dt evaluated from 0 to x, where x is approximately π/2 radians.

Angle1.570796 rad

Solve for:

Hint: The antiderivative of cos(x) is sin(x). Evaluate sin(x) minus sin(0).

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Stewart, James. Calculus: Early Transcendentals.
  2. Halliday, David, Robert Resnick, and Jearl Walker. Fundamentals of Physics.
  3. Wikipedia: Antiderivative
  4. Wikipedia: Trigonometric functions
  5. Atkins' Physical Chemistry, 11th Edition
  6. Halliday, Resnick, and Walker, Fundamentals of Physics, 11th Edition
  7. Wikipedia: Radian
  8. IUPAC Gold Book: radian