Integral Definition:
A numerical value:
Area under the graph of a
function for some interval.
Limit of a Function Notation:
The limit of a function ƒ(x) is L as L approaches a
limx → a ƒ(x) = L
Integral Notation:
∫ƒ(x) dx.
integral of f of x dx.
Integral of a constant rule:
Integral of a constant n isƒ(x) = 6, then ∫ƒ(x) = 6x + C
nx + C.
Integral power rule:
ƒ(x) = xn, then ∫ƒ(x)dx is:
| xn + 1 | |
| n + 1 |
Integral power rule example:
ƒ(x) = x2Using the power rule with n = 2
∫ƒ(x) = x2 + 1/(2 + 1) = x3/3
Trigonometric Integrals
| ƒ(x) | ƒ'(x) | Domain |
|---|---|---|
| sin(x) | -cos(x) | -∞ < x < ∞ |
| cos(x) | sin(x) | -∞ < x < ∞ |
| tan(x) | -Ln cos(x) | x ≠ π/2 + πn, n ∈ Ζ |
| csc(x) | Ln(csc(x) - cot(x)) | x ≠ πn, n ∈ Ζ |
| sec(x) | Ln(sec(x) - tan(x)) | x ≠ π/2 + πn, n ∈ Ζ |
| cot(x) | Ln sin(x) | x ≠ πn, n ∈ Ζ |
Logarithmic Integrals
| ƒ(x) | ƒ'(x) |
|---|---|
| ex | ex |
| ax | ax/Ln(a) where a > 0, a ≠ 1 |
| Ln(x) | 1/x |
| logax | x * Ln(x) - x |
Integral Calculator:
For more help on integralsHow does the Integrals Calculator work?
Free Integrals Calculator - This lesson walks you through the integral definition, rules, and examples
What 3 formulas are used for the Integrals Calculator?
limx → a ƒ(x) = L
ƒ(x) = c, then ∫ƒ(x) = cx + C
ƒ(x) = xn then ƒ'(x) = xn + 1/(n + 1)
ƒ(x) = c, then ∫ƒ(x) = cx + C
ƒ(x) = xn then ƒ'(x) = xn + 1/(n + 1)
What 18 concepts are covered in the Integrals Calculator?
- area
- Number of square units covering the shape
- constant
- a value that always assumes the same value independent of how its parameters are varied
- cos
- cos(θ) is the ratio of the adjacent side of angle θ to the hypotenuse
- cot
- The length of the adjacent side divided by the length of the side opposite the angle. Also equals 1/tan(θ)
- csc
- the length of the hypotenuse divided by the length of the adjacent side. Also equals 1/sin(θ)
- derivative
- rate at which the value y of the function changes with respect to the change of the variable x
- exponent
- The power to raise a number
- function
- relation between a set of inputs and permissible outputs
ƒ(x) - graph
- a pictorial representation or a diagram that represents data or values in an organized manner
- integral
- a mathematical object that can be interpreted as an area or a generalization of area
- interval
- a set of real numbers that contains all real numbers lying between any two numbers of the set
- limit
- the value that a function (or sequence) approaches as the input (or index) approaches some valu
- logarithm
- the exponent or power to which a base must be raised to yield a given number
- power
- how many times to use the number in a multiplication
- sec
- the length of the hypotenuse divided by the length of the adjacent side. Also equals 1/cos(θ)
- sin
- sin(θ) is the ratio of the opposite side of angle θ to the hypotenuse
- tan
- the ratio of the opposite side to the adjacent side of a particular angle of the right triangle.
- variable
- Alphabetic character representing a number
