Definite integral Calculator
Online Definite Integral Calculator
This tool enables online calculation of definite integrals for various functions. It accepts a wide range of functions, including polynomial, exponential, logarithmic, and trigonometric functions.
The Importance of Integrals in Mathematics
Definite integrals play a crucial role in mathematics and in fields such as physics and engineering. They allow for the calculation of quantities such as areas under curves, volumes, and other measures related to rates of change.
How to Use the Calculator
Follow these simple steps to use the integral calculator:
- Enter the variable (e.g., x).
- Input the function to be integrated (e.g., x^3).
- Specify the lower bound (a), which can be a value, a constant, or -inf (for negative infinity).
- Specify the upper bound (b), which can be a value, a constant, or +inf (for positive infinity).
Data Entry Guide
This guide helps you correctly input data into the calculator:
Variables | A function can have one or more variables, but only one main variable. A variable is a single lowercase or uppercase letter. Examples: A function f with one main variable : f(x) = 4*x A function g with one main variable x and a secondary parameter m, g(x) = 4*x*m + x + 1, In this case, enter x in the “main variable” field |
---|---|
Numbers | Use a dot as decimal separator |
Operators |
+ (addition), - (substration), * (multiplication), / (division), ^ (power), For multiply operator, enter a*b not a.b nor ab. Example: 2*x. |
Constants | You may use these constants : pi (approx. 3.14), e (approx. 2,72) Examples: f(x) = pi * x or f(x) = e * (x+1+2*e)^{2} |
Common Functions |
You may use theses functions in the expression of f(x) sqrt(x) (square root), exp(x) (exponential function), log(x) or ln (natural logarithm), |
Trigonometric functions |
You may use theses functions in the expression of f(x) sin (sine), cos (cosine), tan (tangent), cot (cotangent), sec (secant), csc (cosecant), |
Inverse trigonometric functions |
You may use theses functions in the expression of f(x) arcsin (arcsine), arccos (arccosine), arctan (arctangent), arccot (arcotangent), arcsec (arcsecant), arccsc (arccosecant), |
Hyperbolic Functions |
You may use theses functions in the expression of f(x) sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), coth (hyperbolic cotangent), sech (hyperbolic secante), csch (hyperbolic cosecant) |
Inverse hyperbolic functions |
You may use theses functions in the expression of f(x) asinh (inverse hyperbolic sine), acosh (inverse hyperbolic cosine), atanh (inverse hyperbolic tangent), acoth (inverse hyperbolic cotangent), asech (inverse hyperbolic secant), acsch (inverse hyperbolic cosecant) |
Calculating an Integral Over an Interval
To calculate an integral over a given interval, it is essential to know the antiderivative of the function being integrated. The fundamental rule is applying the Fundamental Theorem of Calculus, which states that the definite integral from `a` to `b` of a function `f(x)` is equal to the difference between the values of its antiderivative at points `b` and `a`. In formula, this is written as `F(b) - F(a)`, where `F` is the antiderivative of `f`.
`\int_a^b f(x) \ dx = F(b)-F(a)`
For integrals involving infinity (`+oo` or `-oo`), the approach is to consider these limits as tending towards infinity. For example, to calculate the integral from `a` to `+oo`, evaluate the limit of `F(x)` as `x` tends towards infinity and subtract `F(a)`.
To easily find the antiderivative of a function, you can use our Antiderivative Calculator.
Table of basic functions primitives
Fonction f(x) | Primitive |
---|---|
`k` where `k in RR` | `k*x+C` |
`x^n` where `n in NN`* | `x^(n+1)/(n+1)+C` |
`1/x` | `ln(|x|)+C` |
`1/x^n` where `n in NN , n >=2` | `-1/((n-1)x^(n-1))+C` |
`sqrt(x)` | `frac{2}{3}*x*sqrt(x)+C` |
`1/sqrt(x)` | `-1/(2*x*sqrt(x))+C` |
`sin(x)` | `-cos(x)+C` |
`cos(x)` | `sin(x)+C` |
`ln(x)` | `x*ln(x)-x+C` |
`e^x` | `e^x+C` |
Table of composite functions primitives
Fonction composée | Primitive |
---|---|
`u'*u` | `u^2/2+C` |
`(u')/u^2` | `-1/u+C` |
`u'*u^n` where `n in NN\text{ and }n != -1` | `u^(n+1)/(n+1)+C` |
`(u')/u^n` where `n in NN\text{ and }n >= 2` | `1/((n-1)*u^(n-1))+C` |
`(u')/sqrt(u)` | `2*sqrt(u)+C` |
`(u')/u` | `ln(|u|)+C` |
`u'*e^u` | `e^u+C` |
`u'*sin(u)` | `-cos(u)+C` |
`u'*cos(u)` | `sin(u)+C` |
`u'*u^a` where `a in RR\text{ and }a != -1` | `u^(a+1)/(a+1)+C` |
`u'*g(u)` | `g(u)+C` |
Examples of Integral Calculation
These examples illustrate how to use the calculator for different functions and also demonstrate the manual calculation method for each type of integral:
Integral of a Polynomial Function
Using the Calculator: To calculate the integral of `f(x) = x^3` over the interval from 1 to 3, enter x as the variable, x^3 as the function, 1 for the lower bound, and 3 for the upper bound.
Manual Method: The integral of `x^3` is `(x^4)/4`. Over the interval from 1 to 3, calculate `(3^4)/4 - (1^4)/4` to obtain the result which is equal to 20.
Integral of a Trigonometric Function
Using the Calculator: To integrate `f(x) = sin(x)` from 0 to pi, enter x as the variable, sin(x) as the function, 0 for the lower bound, and pi for the upper bound.
Manual Method: The integral of `sin(x)` is `-cos(x)`. Apply the limit from 0 to pi to get `-cos(pi) - (-cos(0))`, which is equal to 2.
Integral of a Logarithmic Function
Using the Calculator: For the integral of `f(x) = ln(x)` from 1 to e, enter x as the variable, ln(x) as the function, 1 for the lower bound, and e for the upper bound.
Manual Method: The integral of `ln(x)` is `x*ln(x) - x`. Applying the limits from 1 to e, the calculation becomes `e*ln(e) - e - (1*ln(1) - 1)`, which simplifies to 1.
Conclusion
This online definite integral calculator is a powerful tool, designed to simplify integration calculations. Whether you are a student, teacher, or professional, it helps you calculate integrals quickly and accurately over various intervals. Explore its capabilities and enhance your understanding of fundamental mathematical concepts.
See also
Derivative of a function
Primitive calculator
Taylor series expansion
Function limit
Value of a function