MathematicsCalculusA-Level

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Integral of cos(x)

Antiderivative of the cosine function.

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\int cos x d x = sin x + C

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Core idea

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.

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.

Symbols

Variables

I = Integral Value, x = Angle

I

Integral Value

(i g n or in g C)

x

Angle

r a d

Walkthrough

Derivation

Formula: Integral of cos(x)

The integral of cos(x) is sin(x), reversing the differentiation result for sine.

x is measured in radians.
Integration is with respect to x.

Recall the Derivative of Sine:

Differentiating sin gives cos.

\frac{d}{d x} (sin x) = cos x

State the Integral:

Reverse the differentiation result and add the constant of integration.

\int cos x d x = sin x + C

Note: Common sign errors happen with trig calculus; cos integrates to +sin.

Result

\int cos x d x = sin x + C

Source: OCR A-Level Mathematics — Pure (Integration)

Visual intuition

Graph

The graph is a sinusoidal wave plotted with the independent variable on the x-axis and the integral value on the y-axis. This periodic shape occurs because the integral of the cosine function results in the sine function, which oscillates continuously between -1 and 1.

Graph type: sinusoidal

Why it behaves this way

Intuition

The integral of cos x visualizes finding a curve (sin x) whose instantaneous slope at any point x is given by the value of cos x at that point.

\int

The operation of integration, representing the accumulation of infinitesimal quantities or finding the antiderivative.

It signifies summing up tiny slices of the function's value to find the total change or area under the curve.

cos x

The instantaneous rate of change or velocity of a sinusoidally oscillating system at a given point 'x'.

It describes an oscillation that starts at its peak (for x=0) and cycles smoothly, indicating how fast and in what direction a quantity is changing.

An infinitesimally small increment of the independent variable 'x'.

It represents the 'width' of each tiny slice of the function being summed up during integration.

sin x

The antiderivative of cos x, representing the position or accumulated quantity of a sinusoidally oscillating system whose rate of change is cos x.

It describes an oscillation that starts at zero (for x=0) and cycles smoothly, representing the total amount or position reached given the cos x rate of change.

The constant of integration, representing an arbitrary vertical shift of the antiderivative.

Since the derivative of any constant is zero, 'C' accounts for the unknown initial condition or starting point of the original function before it was differentiated.

Free study cues

Insight

Canonical usage

The integral of a dimensionless trigonometric function cos(x) with respect to x results in a quantity having the same dimensions as x.

Common confusion

A common mistake is forgetting that the variable x inside cos(x) is assumed to be in radians for the standard integral formula to hold.

Dimension note

While the trigonometric functions cos(x) and sin(x) are themselves dimensionless, the integral ∫ cos x dx takes on the dimension of the integration variable x.

Unit systems

$x$ radians (dimensionless) or any base unit (e.g., s, m) · In calculus, x is typically an angle in radians for the derivative/integral formulas to be valid without scaling factors. If x represents a physical quantity (e.g., time, length), its units and dimension will determine

$cos x$ dimensionless · Trigonometric functions are inherently dimensionless ratios.

$s in x$ dimensionless · Trigonometric functions are inherently dimensionless ratios.

$C$ same as x · The constant of integration must have the same dimensions as the integral itself, which are the dimensions of x.

Ballpark figures

Quantity:

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: $I$

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

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Signal processing.

Study smarter

Tips

Always remember that the integral of cosine is positive sine, while the derivative is negative sine.
Ensure your calculator is in radians mode, as calculus operations with trig functions rely on radian measure.
Include the constant of integration C for indefinite integrals to account for all possible vertical shifts.

Avoid these traps

Common Mistakes

Adding negative sign.
Using degrees.

Common questions

Frequently Asked Questions

The integral of cos(x) is sin(x), reversing the differentiation result for sine.

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.

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.

Adding negative sign. Using degrees.