Given ƒ(x) = 3x4 + 6x3 - 123x2 - 126x + 1080dx
Determine the integral ∫ƒ(x)
Go through and integrate each term
Integrate term 1
ƒ(x) = 3x4Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= 3, n = 4
and x is the variable we integrate
| ∫ƒ(x) = | 3x(4 + 1) |
| 4 + 1 |
| ∫ƒ(x) = | 3x5 |
| 5 |
Integrate term 2
ƒ(x) = 6x3Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= 6, n = 3
and x is the variable we integrate
| ∫ƒ(x) = | 6x(3 + 1) |
| 3 + 1 |
| ∫ƒ(x) = | 6x4 |
| 4 |
Simplify our fraction.
Divide top and bottom by 2
| ∫ƒ(x) = | + 3x4 |
| 2 |
Integrate term 3
ƒ(x) = -123x2Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= -123, n = 2
and x is the variable we integrate
| ∫ƒ(x) = | -123x(2 + 1) |
| 2 + 1 |
| ∫ƒ(x) = | -123x3 |
| 3 |
Simplify our fraction.
Divide top and bottom by 3
∫ƒ(x) = -41x3
Integrate term 4
ƒ(x) = -126xUse the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= -126, n = 1
and x is the variable we integrate
| ∫ƒ(x) = | -126x(1 + 1) |
| 1 + 1 |
| ∫ƒ(x) = | -126x2 |
| 2 |
Simplify our fraction.
Divide top and bottom by 2
∫ƒ(x) = -63x2
Integrate term 5
ƒ(x) = 1080Use the power rule
∫ƒ(x) of the expression axn| ax(n + 1) | |
| n + 1 |
= 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) = 978.1
Evaluate ∫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) = 0
Determine 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
