Given ƒ(x) = 6x2 + 4x - 1dx
Determine the integral ∫ƒ(x)
Go through and integrate each term
Integrate term 1
ƒ(x) = 6x2Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= 6, n = 2
and x is the variable we integrate
| ∫ƒ(x) = | 6x(2 + 1) |
| 2 + 1 |
| ∫ƒ(x) = | 6x3 |
| 3 |
Simplify our fraction.
Divide top and bottom by 3
∫ƒ(x) = 2x3
Integrate term 2
ƒ(x) = 4xUse the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= 4, n = 1
and x is the variable we integrate
| ∫ƒ(x) = | 4x(1 + 1) |
| 1 + 1 |
| ∫ƒ(x) = | 4x2 |
| 2 |
Simplify our fraction.
Divide top and bottom by 2
∫ƒ(x) = 2x2
Integrate term 3
ƒ(x) = -1Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= -1, n = 0
and x is the variable we integrate
| ∫ƒ(x) = | -1x(0 + 1) |
| 0 + 1 |
∫ƒ(x) = -x
Collecting all of our integrated terms we get:
∫ƒ(x) = 2x3 + 2x2 - xEvaluate ∫ƒ(x) on the interval [0,1]
The value of the integral over an interval is ∫ƒ(1) - ∫ƒ(0)Evaluate ∫ƒ(1)
∫ƒ(1) = 2(1)3 + 2(1)2 - (1)∫ƒ(1) = 2(1) + 2(1) - (1)
∫ƒ(1) = 2 + 2 - 1
∫ƒ(1) = 3
Evaluate ∫ƒ(0)
∫ƒ(0) = 2(0)3 + 2(0)2 - (0)∫ƒ(0) = 2(0) + 2(0) - (0)
∫ƒ(0) = 0 - 0 - 0
∫ƒ(0) = 0
Determine our answer
∫ƒ(x) on the interval [0,1] = ∫ƒ(1) - ∫ƒ(0)∫ƒ(x) on the interval [0,1] = 3 - 0
Final Answer
∫ƒ(x) on the interval [0,1] = 3
You have 2 free calculationss remaining
What is the Answer?
∫ƒ(x) on the interval [0,1] = 3
How does the Functions-Derivatives-Integrals Calculator work?
Free Functions-Derivatives-Integrals Calculator - Given a polynomial expression, this calculator evaluates the following items:
1) Functions ƒ(x). Your expression will also be evaluated at a point, i.e., ƒ(1)
2) 1st Derivative ƒ‘(x) The derivative of your expression will also be evaluated at a point, i.e., ƒ‘(1)
3) 2nd Derivative ƒ‘‘(x) The second derivative of your expression will be also evaluated at a point, i.e., ƒ‘‘(1)
4) Integrals ∫ƒ(x) The integral of your expression will also be evaluated on an interval, i.e., [0,1]
5) Using Simpsons Rule, the calculator will estimate the value of ≈ ∫ƒ(x) over an interval, i.e., [0,1]
This calculator has 7 inputs.
1) Functions ƒ(x). Your expression will also be evaluated at a point, i.e., ƒ(1)
2) 1st Derivative ƒ‘(x) The derivative of your expression will also be evaluated at a point, i.e., ƒ‘(1)
3) 2nd Derivative ƒ‘‘(x) The second derivative of your expression will be also evaluated at a point, i.e., ƒ‘‘(1)
4) Integrals ∫ƒ(x) The integral of your expression will also be evaluated on an interval, i.e., [0,1]
5) Using Simpsons Rule, the calculator will estimate the value of ≈ ∫ƒ(x) over an interval, i.e., [0,1]
This calculator has 7 inputs.
What 1 formula is used for the Functions-Derivatives-Integrals Calculator?
What 8 concepts are covered in the Functions-Derivatives-Integrals Calculator?
- 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) - functions-derivatives-integrals
- integral
- a mathematical object that can be interpreted as an area or a generalization of area
- point
- an exact location in the space, and has no length, width, or thickness
- polynomial
- an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
- power
- how many times to use the number in a multiplication
Example calculations for the Functions-Derivatives-Integrals Calculator
Tags:
Add This Calculator To Your Website
