Given ƒ(x) = 5x3dx
Determine the integral ∫ƒ(x)
Go through and integrate each term
Integrate term 1
ƒ(x) = 5x3Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= 5, n = 3
and x is the variable we integrate
| ∫ƒ(x) = | 5x(3 + 1) |
| 3 + 1 |
| ∫ƒ(x) = | 5x4 |
| 4 |
Collecting all of our integrated terms we get:
∫ƒ(x) = 5x4/4Evaluate ∫ƒ(x) on the interval [0,]
The value of the integral over an interval is ∫ƒ() - ∫ƒ(0)Evaluate ∫ƒ()
∫ƒ() = 5()4/4∫ƒ() = 5(0)/4
∫ƒ() = 0
∫ƒ() = 0
Evaluate ∫ƒ(0)
∫ƒ(0) = 5(0)4/4∫ƒ(0) = 5(0)/4
∫ƒ(0) = 0
∫ƒ(0) = 0
Determine our answer
∫ƒ(x) on the interval [0,] = ∫ƒ() - ∫ƒ(0)∫ƒ(x) on the interval [0,] = 0 - 0
Final Answer
∫ƒ(x) on the interval [0,] = 0
What is the Answer?
∫ƒ(x) on the interval [0,] = 0
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?
Power Rule:
f(x) = xn, f‘(x) = nx(n - 1)
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
