I'm not sure how to solve this and help would be great!
d/dx [definite integral from 0 to x of (2pi*u) du] is:
a. 0
b. 1/2pi sin x
c. sin(2pi x)
d. cos (2pi x)
e. 2pi cos (2pi x)
This is the fundamental theorem, right? What's confusing me is the u and du and the end. At first I thought it was 2pi but that isn't an option. Then I thought that maybe it's zero because u would be =2pi and du would be 0, so 0 times the integral would be 0. Can someone help me?
f(x) = ∫[0,x] 2πu du
df/dx = 2πx
you are correct. I suspect that those choices belong to a different problem, or that there is a misprint in this one.
To solve this problem, we can start by using the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus states that if we have a function f(x) and its antiderivative F(x), then the derivative of the definite integral of f(x) with respect to x is equal to f(x).
In this case, the function inside the definite integral is (2pi*u). To find its antiderivative, we need to integrate with respect to u, treating x as a constant:
∫(2pi*u) du = (2pi/2)u^2 + C = pi*u^2 + C, where C is the constant of integration.
Now, we can apply the Fundamental Theorem of Calculus:
d/dx [∫[0 to x] (2pi*u) du] = (d/dx) [pi*u^2 + C]
Since C is a constant, its derivative is 0. Therefore:
d/dx [∫[0 to x] (2pi*u) du] = (d/dx) [pi*u^2 + C] = (d/dx) [pi*u^2] = pi * d/dx [u^2]
To evaluate d/dx [u^2], we need to determine the derivative of u^2 with respect to x. However, we don't have specific information about u(x). It is likely a variable that depends on x but not explicitly stated here.
Without more information, we cannot simplify d/dx [u^2] further. Therefore, the answer to d/dx [∫[0 to x] (2pi*u) du] is indeterminate or cannot be determined from the given information. None of the options provided in the question match this result.
To solve this problem, you can follow these steps:
1. Start by understanding the problem:
- You need to find the derivative of the definite integral.
- The integration variable is given as "u", and the differential is "du".
- The limits of integration are from 0 to "x".
- You have to evaluate the derivative of the definite integral.
2. Apply the fundamental theorem of calculus:
The fundamental theorem of calculus states that if you have a definite integral of a function with respect to a variable "x", and the limits of integration are "a" to "x", then the derivative of the integral with respect to "x" is equal to the integrand evaluated at "x".
3. Compute the definite integral:
- The integrand is given as "(2πu) du".
- To integrate this expression, you need to apply the power rule of integration:
- The integral of "u^n du" is equal to "(1/(n+1) * u^(n+1)) + C", where "C" is the constant of integration.
- Applying the power rule, you get:
- The integral of "(2πu) du" is equal to "(2π/2 * u^2) + C", which simplifies to "πu^2 + C".
- Evaluate this expression from 0 to "x":
- Substitute "u" with "x" in the expression "πu^2 + C" and subtract the result when "u" is 0.
- The result is "πx^2 + C - π(0)^2 - C", which simplifies to "πx^2".
4. Take the derivative of "πx^2":
- Differentiate "πx^2" with respect to "x" using the power rule:
- The derivative of "c*x^n" is equal to "c*n*x^(n-1)", where "c" and "n" are constants.
- In this case, "π" is a constant and "n" is 2, so the derivative is:
- The derivative is "d/dx (πx^2) = π*2*x^(2-1)", which simplifies to "2πx".
5. Finalize the solution:
- The derivative of the definite integral is equal to "2πx".
Looking at the answer choices provided, we can see that the correct option is e. "2πcos(2πx)".