Given ƒ(x) = x2 - x - 12dx
Determine the integral ∫ƒ(x)
Go through and integrate each term
itarget = -5,10
Integrate term 1
ƒ(x) = x2Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= 1, n = 2
and x is the variable we integrate
| ∫ƒ(x) = | 1x(2 + 1) |
| 2 + 1 |
| ∫ƒ(x) = | 1x3 |
| 3 |
Integrate term 2
ƒ(x) = -xUse the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= -1, n = 1
and x is the variable we integrate
| ∫ƒ(x) = | -1x(1 + 1) |
| 1 + 1 |
| ∫ƒ(x) = | -1x2 |
| 2 |
Integrate term 3
ƒ(x) = -12Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= -12, n = 0
and x is the variable we integrate
| ∫ƒ(x) = | -12x(0 + 1) |
| 0 + 1 |
∫ƒ(x) = -12x
Collecting all of our integrated terms we get:
∫ƒ(x) = x3/3 - x2/2 - 12xEvaluate ∫ƒ(x) on the interval [-5,10]
The value of the integral over an interval is ∫ƒ(10) - ∫ƒ(-5)Evaluate ∫ƒ(10)
∫ƒ(10) = (10)3/3 - (10)2/2 - 12(10)∫ƒ(10) = (1000)/3 - (100)/2 - 12(10)
∫ƒ(10) = 333.33333333333 - 50 - 120
∫ƒ(10) = 163.33333333333
Evaluate ∫ƒ(-5)
∫ƒ(-5) = (-5)3/3 - (-5)2/2 - 12(-5)∫ƒ(-5) = (-125)/3 - (25)/2 - 12(-5)
∫ƒ(-5) = -41.666666666667 - 12.5 + 60
∫ƒ(-5) = 5.8333333333333
Determine our answer
∫ƒ(x) on the interval [-5,10] = ∫ƒ(10) - ∫ƒ(-5)∫ƒ(x) on the interval [-5,10] = 163.33333333333 - 5.8333333333333
Final Answer
∫ƒ(x) on the interval [-5,10] = 157.5
You have 2 free calculationss remaining
What is the Answer?
∫ƒ(x) on the interval [-5,10] = 157.5
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
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