Integral of sin(x) Calculator
Antiderivative of the sine function.
Formula first
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.
Symbols
Variables
I = Integral Value, x = Angle, x_u = Upper Limit, x_l = Lower Limit, I_{def} = Definite Integral Value
Apply it well
When To Use
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.
Avoid these traps
Common Mistakes
- Omitting negative sign.
- Mixing differentiation and integration.
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.
Solve for:
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.
References
Sources
- Stewart, James. Calculus: Early Transcendentals. Cengage Learning.
- Wikipedia: Antiderivative
- Halliday, Resnick, Walker, Fundamentals of Physics
- Atkins' Physical Chemistry
- Wikipedia: Radian
- Wikipedia: Trigonometric functions
- Stewart, James. Calculus: Early Transcendentals, 8th Edition.
- Thomas' Calculus, 14th Edition.