Integral of sin(x) Equation, Derivation & Rearrangements

Core idea

Overview

The integral of the sine function identifies the antiderivative that, when differentiated, produces the original sine wave. This mathematical operation results in the negative cosine function, which is critical for solving problems involving cyclical and oscillatory systems.

When to use: Apply this formula when you need to calculate the area under a sine curve or determine the accumulation of a quantity varying sinusoidally over time. It is specifically used in kinematics to find position when velocity is described as a sine function or in electricity to find average values of alternating current.

Why it matters: This integral is fundamental for describing physical phenomena such as sound waves, light waves, and harmonic motion. It provides the essential mathematical link between orthogonal trigonometric components and their dynamic behavior in physics and engineering applications.

Symbols

Variables

I = Integral Value, x = Angle, x_u = Upper Limit, x_l = Lower Limit, I_{def} = Definite Integral Value

I

Integral Value

(i g n or in g C)

x

Angle

r a d

x_{u}

Upper Limit

r a d

x_{l}

Lower Limit

r a d

I_{d e f}

Definite Integral Value

(f r o m l o w er t o u pp er l imi t)

Walkthrough

Derivation

Formula: Integral of sin(x)

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

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

1

Recall the Derivative of Cosine:

Differentiating cos gives negative sin.

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

2

Adjust the Sign:

So an antiderivative of $sin x$ is $- cos x$ .

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

3

State the Integral:

Include the constant of integration C for an indefinite integral.

\int sin x d x = - cos x + C

Result

\int sin x d x = - cos x + C

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

Visual intuition

Graph

The graph follows a smooth sinusoidal oscillation between negative one and one because the result is defined by the negative cosine of the variable. For a student, this shape demonstrates that the accumulated area under the sine curve repeats periodically, meaning large values of x do not lead to infinite growth but rather cycle through the same range of values. The most important feature is that the curve crosses the horizontal axis at regular intervals, which represents points where the total accumulated area of

Graph type: sinusoidal

Why it behaves this way

Intuition

Imagine the integral as continuously summing the heights of the sine wave over tiny intervals, resulting in a new wave (negative cosine)

\int

The integral operator, representing the process of finding the antiderivative or the accumulation of a function's values.

Imagine summing up infinitely many tiny vertical slices of the function's height to find the total area under its curve.

sin x

The integrand, the function whose antiderivative is being sought. It represents a sinusoidal oscillation.

This is the 'input' wave, varying smoothly between -1 and 1, whose accumulated effect we are measuring.

dx

The differential element, indicating that the integration is with respect to the variable x and representing an infinitesimally small increment along the x-axis.

The tiny, vanishingly small width of each slice under the curve that we are adding up.

- cos x

The antiderivative of sin x, meaning the function that, when differentiated, yields sin x.

This is the 'output' wave, a shifted and inverted cosine curve, representing the total accumulated value of sin x up to any point.

+ C

The constant of integration, representing an arbitrary constant value that vanishes upon differentiation. It accounts for the family of antiderivatives.

Since differentiating a constant results in zero, there's an infinite set of possible antiderivatives, all vertically shifted versions of each other.

Signs and relationships

-\cos x: The negative sign is crucial because the derivative of cos x is -sin x. Therefore, to obtain a positive sin x from differentiation, the antiderivative must be -cos x, as d/dx(-cos x) = -(-sin x) = sin x.

Free study cues

Insight

Canonical usage

In pure mathematics and physics, the argument x is treated as a dimensionless quantity (typically in radians), making the integral and its result also dimensionless.

Common confusion

Students often forget that the argument of trigonometric functions must be dimensionless. If x represents a physical quantity with units, they might incorrectly apply the formula without a necessary scaling factor (e.g.

Dimension note

The argument x of the sine function is inherently dimensionless (e.g., an angle in radians). Consequently, sin x and cos x are dimensionless.

Unit systems

$x$ rad · The argument x of trigonometric functions must be dimensionless. In the context of this integral, x is typically treated as an angle in radians.

One free problem

Practice Problem

Evaluate the definite integral of sin(x) from a lower limit of 0 to an upper limit of x = 3.14159.

Upper Limit3.14159 rad

Lower Limit0 rad

Solve for: $I_{d} e f ini t e$

Hint: Evaluate the expression -cos(x) at the upper bound and subtract the value at the lower bound.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Average value of AC current.

Study smarter

Tips

Always remember the negative sign: the integral of sine is negative cosine.
Check results by differentiating back to the original sine function.
Remember the constant of integration C for all indefinite integrals.
Ensure the variable x is in radians before evaluating the cosine function.

Avoid these traps

Common Mistakes

Omitting negative sign.
Mixing differentiation and integration.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

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

Apply this formula when you need to calculate the area under a sine curve or determine the accumulation of a quantity varying sinusoidally over time. It is specifically used in kinematics to find position when velocity is described as a sine function or in electricity to find average values of alternating current.

This integral is fundamental for describing physical phenomena such as sound waves, light waves, and harmonic motion. It provides the essential mathematical link between orthogonal trigonometric components and their dynamic behavior in physics and engineering applications.

Omitting negative sign. Mixing differentiation and integration.