Integral Calculator
Find the indefinite integral (antiderivative) of common functions.
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Understanding Integrals
Integration is the inverse process of differentiation. It is used to find the area under a curve, the volume of a solid, and in many other applications involving accumulation.
Basic Integration Rules:
- Power Rule: ∫xn dx = (xn+1 / (n+1)) + C (for n ≠ -1)
- Constant Rule: ∫c dx = cx + C
- Constant Multiple Rule: ∫cf(x) dx = c ∫f(x) dx
- Sum/Difference Rule: ∫(f(x) ± g(x)) dx = ∫f(x) dx ± ∫g(x) dx
- Integral of sin(x): ∫sin(x) dx = -cos(x) + C
- Integral of cos(x): ∫cos(x) dx = sin(x) + C
- Integral of 1/x: ∫(1/x) dx = ln|x| + C
Table of Common Integrals
| Function f(x) | Integral ∫f(x) dx |
|---|---|
| c (constant) | cx + C |
| xn (n ≠ -1) | xn+1 / (n+1) + C |
| 1/x | ln|x| + C |
| ex | ex + C |
| sin(x) | -cos(x) + C |
| cos(x) | sin(x) + C |
| sec2(x) | tan(x) + C |
Frequently Asked Questions
What is an integral?
In calculus, an integral is a mathematical object that can be interpreted as an area, a generalization of mass, or the total change of a quantity given its rate of change. The process of finding integrals is called integration.
What is the difference between indefinite and definite integrals?
An indefinite integral (or antiderivative) is a function whose derivative is the original function, and it includes an arbitrary constant of integration (+ C). A definite integral, on the other hand, evaluates the integral over a specific interval and represents the net change or area under the curve between two points, resulting in a numerical value.
What are integrals used for?
Integrals are used to calculate areas under curves, volumes of solids, lengths of curves, and in physics to find displacement from velocity, work done by a force, and many other applications involving accumulation or total change.