Find the derivative of the following function using the appropriate form of the Fundamental Theorem of Calculus.
intergral s^2/(1+3s^4) ds from sqrtx to 1
F'(x)=?
Find the most general antiderivative of f(x)=–8e^x–6secant^2(x), where -pi/2<x<pi/2
Note: Any arbitrary constants used must be an upper-case "C"
F(x)=?
∫[√x,1] s^2/(1+3s^4) ds
Are you sure there's no typo here? As you can see here
http://www.wolframalpha.com/input/?i=%E2%88%ABs^2%2F%281%2B3s^4%29+ds
this is not an integral I'd expect to find. If you meant
∫[√x,1] s^3/(1+3s^4) ds
then it's a lot easier:
1/12 log(1+3s^4) [√x,1]
= 1/12 (log(1+3x^2)-log(4))
= 1/12 log((1+3x^2)/4)
To find the derivative of the given function using the Fundamental Theorem of Calculus, we will use the second part of the theorem, which states that if F(x) is the integral of f(t) from a to x, then F'(x) = f(x).
In this case, our integral is ∫(s^2 / (1+3s^4)) ds, and we want to find F'(x) for the limits of integration from √x to 1.
Step 1: Evaluate the integral ∫(s^2 / (1+3s^4)) ds
The integral of a rational function like this can be found by using partial fraction decomposition and integrating each term individually.
Step 2: Perform partial fraction decomposition
To do this, we factorize the denominator as much as possible: 1+3s^4 = (1+s^2)(1-s^2√3)(1+s^2√3).
Then, we set up the partial fraction decomposition:
s^2 / (1+3s^4) = A / (1+s^2) + B / (1-s^2√3) + C / (1+s^2√3)
We need to solve for the values of A, B, and C.
Step 3: Solve for A, B, and C
To solve for A, we multiply both sides of the equation by (1+s^2):
s^2 = A + B(1+s^2√3) + C(1-s^2√3)
By substituting s = 0, we can solve for A:
0 = A + B + C
To solve for B, we multiply both sides of the equation by (1-s^2√3):
s^2(1+s^2√3) = A(1-s^2√3) + B(1+s^2√3)(1+s^2√3) + C(1-s^2√3)(1+s^2√3)
Expanding simplifies the equation to:
s^2 + s^4√3 = A - As^2√3 + B + Bs^4√3 + C - Cs^2√3
By substituting s = 1/√3, we can solve for B:
1/3 + 1/3 = A - A/√3 + B + B/3 + C - C/√3
2/3 = A/√3 + (4/3)B/√3 + C/√3
Now, we solve for C by substituting s = -1/√3:
1/3 + 1/3 = A + A/√3 + B - B/3 + C + C/√3
2/3 = A/√3 - (2/3)B/√3 + C/√3
We have two equations:
0 = A + B + C
2/3 = A/√3 + (4/3)B/√3 + C/√3
We can solve this system of equations to find A, B, and C. Solving this system of equations might involve some algebraic manipulation involving square roots.
Step 4: Integrate each term individually
Once you have found the values of A, B, and C, integrate each term individually:
∫(s^2 / (1+3s^4)) ds = ∫[ A / (1+s^2) + B / (1-s^2√3) + C / (1+s^2√3) ] ds
The integrals of each term can be evaluated using basic integral rules and techniques.
Step 5: Evaluate the definite integral
After integrating each term, evaluate the definite integral from √x to 1:
F(x) = ∫[ A / (1+s^2) + B / (1-s^2√3) + C / (1+s^2√3) ] ds
F(1) - F(√x)
Step 6: Calculate F'(x)
Finally, differentiate F(x) by finding the derivative of each term with respect to x:
F'(x) = d/dx [ ∫[ A / (1+s^2) + B / (1-s^2√3) + C / (1+s^2√3) ] ds ]
Take the derivative of each term using the chain rule and evaluate the result.
This is the process to find the derivative of the given function using the Fundamental Theorem of Calculus with appropriate partial fraction decomposition. It involves solving for the constants A, B, and C and integrating each term individually before evaluating the definite integral and differentiating it.