Consider the functions: F(x)= definite integral [1,x] of f(t)dt f(t)= definite integral [1,t^2] of (sqrt(5+u^4)/u) du Find F''(1). Use the Fundamental Theorem of Calculus
- print Print
- list Cite
Expert Answers
calendarEducator since 2011
write5,349 answers
starTop subjects are Math, Science, and Business
You need to evaluate f(t), hence, you need to solve the definite integral `int_1^(t^2)(sqrt(5+u^4))/u du` such that:
`u^4 =v => 4u^3du = dv => du = (dv)/(4u^3)`
`int (sqrt(5+u^4))/u du = int (sqrt(5+v))/u(dv)/(4u^3) `
`int (sqrt(5+v))/(4u^4) dv = (1/4)int (sqrt(5+v))/v(dv)`
You should consider the next substitution such that:
`sqrt(5+v) = s => s^2 = 5+v => v = s^2-5`
`1/(2sqrt(5+v))dv = ds => dv = 2sqrt(5+v)ds => dv = 2s ds`
`(1/4)int (sqrt(5+v))/v(dv) = (1/2)int s^2/(s^2-5)ds`
You need to subtract and add 5 to numerator such that:
`int (s^2-5+5)/(s^2-5)ds `
Using the property of linearity yields:
`int (s^2-5+5)/(s^2-5)ds = int (s^2-5)/(s^2-5)ds + int (5)/(s^2-5)ds `
`int (s^2-5+5)/(s^2-5)ds = int ds + 5 ln|(s-5)/(s+5)| + c`
`(1/2)int s^2/(s^2-5)ds = s/2 + (5/2)ln|(s-5)/(s+5)| + c`
Substituting back `sqrt(5+v)` for s yields:
`(1/4)int (sqrt(5+v))/v(dv) = (sqrt(5+v))/2 + (5/2)ln|(sqrt(5+v)-5)/(sqrt(5+v)+5)| + c`
Substituting back `u^4` for v yields:
`int (sqrt(5+u^4))/u du = (sqrt(5+u^4))/2 + (5/2)ln|(sqrt(5+u^4)-5)/(sqrt(5+u^4)+5)| + c `
Using the fundamental theorem of calculus yields:
`int_1^(t^2) (sqrt(5+u^4))/u du = ((sqrt(5+u^4))/2 + (5/2)ln|(sqrt(5+u^4)-5)/(sqrt(5+u^4)+5)|)|_1^(t^2)`
`int_1^(t^2) (sqrt(5+u^4))/u du = ((sqrt(5+t^8))/2 + (5/2)ln|(sqrt(5+t^8)-5)/(sqrt(5+t^8)+5)| - sqrt(6))/2- (5/2)ln|(sqrt(6)-5)/(sqrt(6)+5))`
Hence, evaluating f(t) yields`f(t) = ((sqrt(5+t^8))/2 + (5/2)ln|(sqrt(5+t^8)-5)/(sqrt(5+t^8)+5)| - sqrt(6))/2 - (5/2)ln|(sqrt(6)-5)/(sqrt(6)+5)).`
You need to remember that `F'(x) = f(x)` and `F''(x) = f'(x), ` hence, you need to evaluate f'(1) to find F''(1) such that:
`f'(t) = (((sqrt(5+t^8))/2)' + (5/2)(ln|(sqrt(5+t^8)-5)| - ln|(sqrt(5+t^8)+5)|)'`
`f'(t) = 2t^7/(sqrt(5+t^8)) + 10(1/(sqrt(5+t^8)-5))(t^7)/(sqrt(5+t^8))- 10(1/(sqrt(5+t^8)+5))(t^7)/(sqrt(5+t^8))`
You need to evaluate f'(1) such that:
`f'(1) = 2/(sqrt 6) + 10(1/(sqrt(6)-5))(1/(sqrt(6))) - 10(1/(sqrt(6)+5))(1/(sqrt(6)))`
`f'(1) = 2/(sqrt 6) + 10/(sqrt 6)(1/(sqrt 6 - 5) - 1/(sqrt 6 + 5))`
`f'(1) = 2/(sqrt 6) + 10/(sqrt 6)(sqrt6 + 5 - sqrt6 + 5)/(6-25)`
`f'(1) = 2/(sqrt 6)(1- 50/19) => f'(1) = 2/(sqrt 6)(-31/19)`
`f'(1) = -62/(19sqrt6) => f'(1) = -62sqrt6/(19*6)`
`f'(1) = -31sqrt6/(19*3) => f'(1) = -31sqrt6/57 = F''(1)`
Hence, evaluating `F''(1)` under the given conditions yields `F''(1) = -31sqrt6/57.`
Related Questions
- Find the indefinite integral `int x sqrt((2x-1)) dx` using integration by substitution `int x...
- 1 Educator Answer
- Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y=...
- 1 Educator Answer
- `int (dt)/(t^2 sqrt(t^2 - 16))` Evaluate the integral
- 2 Educator Answers
- `int u sqrt(1 - u^2) du` Evaluate the indefinite integral.
- 1 Educator Answer
- Integrate f(x) = 1/(x^2 - 4)
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.