Given ƒ(x) = 3x
4 + 6x
3 - 123x
2 - 126x + 1080dx
Determine the integral ∫ƒ(x)
Go through and integrate each term
itarget = 0,1
Integrate term 1
ƒ(x) = 3x
4Use the power rule
∫ƒ(x) of the expression ax
n = 3, n = 4
and x is the variable we integrate
Integrate term 2
ƒ(x) = 6x
3Use the power rule
∫ƒ(x) of the expression ax
n = 6, n = 3
and x is the variable we integrate
Simplify our fraction.
Divide top and bottom by 2
Integrate term 3
ƒ(x) = -123x
2Use the power rule
∫ƒ(x) of the expression ax
n = -123, n = 2
and x is the variable we integrate
∫ƒ(x) = | -123x(2 + 1) |
| 2 + 1 |
Simplify our fraction.
Divide top and bottom by 3
∫ƒ(x) = -41x
3
Integrate term 4
ƒ(x) = -126x
Use the power rule
∫ƒ(x) of the expression ax
n = -126, n = 1
and x is the variable we integrate
∫ƒ(x) = | -126x(1 + 1) |
| 1 + 1 |
Simplify our fraction.
Divide top and bottom by 2
∫ƒ(x) = -63x
2
Integrate term 5
ƒ(x) = 1080
Use the power rule
∫ƒ(x) of the expression ax
n = 1080, n = 0
and x is the variable we integrate
∫ƒ(x) = | 1080x(0 + 1) |
| 0 + 1 |
∫ƒ(x) = 1080x
Collecting all of our integrated terms we get:
∫ƒ(x) =
3x5/5 + 3x4/2 - 41x3 - 63x2 + 1080xEvaluate ∫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) = 3(
1)
5/5 + 3(
1)
4/2 - 41(
1)
3 - 63(
1)
2 + 1080(
1)
∫ƒ(1) = 3(1)/5 + 3(1)/2 - 41(1) - 63(1) + 1080(
1)
∫ƒ(1) = 0.6 + 1.5 - 41 - 63 + 1080
∫ƒ(1) =
978.1Evaluate ∫ƒ(0)
∫ƒ(0) = 3(
0)
5/5 + 3(
0)
4/2 - 41(
0)
3 - 63(
0)
2 + 1080(
0)
∫ƒ(0) = 3(0)/5 + 3(0)/2 - 41(0) - 63(0) + 1080(
0)
∫ƒ(0) = 0 - 0 - 0 - 0 - 0
∫ƒ(0) =
0Determine our answer
∫ƒ(x) on the interval [0,1] = ∫ƒ(1) - ∫ƒ(0)
∫ƒ(x) on the interval [0,1] = 978.1 - 0
Final Answer
∫ƒ(x) on the interval [0,1] = 978.1
You have 2 free calculationss remaining
What is the Answer?
∫ƒ(x) on the interval [0,1] = 978.1
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.
What 1 formula is used for the Functions-Derivatives-Integrals Calculator?
Power Rule:
f(x) = x
n, f‘(x) = nx
(n - 1)For more math formulas, check out our
Formula Dossier
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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