What is the derivative of the integral ∫ from 0 to x of t^2 dt with respect to x?

What is the derivative of the integral ∫ from 0 to x of t^2 dt with respect to x?

• A. x^2
• B. 2x
• C. 3x^2
• D. 1

Answer: (A)

To find the derivative of the integral ∫ from 0 to x of t² dt with respect to x, we can apply the Fundamental Theorem of Calculus. This theorem states that if F(x) is an antiderivative of f(t) on an interval [a, b], then the derivative of the integral from a to x of f(t) dt is simply f(x). In this case, the function we are integrating is t². According to the Fundamental Theorem of Calculus, the derivative of the integral with respect to the upper limit (in this case x) is just the integrand evaluated at x. Therefore, we replace t with x in the function t²: f(x) = x². Thus, the derivative of the integral ∫ from 0 to x of t² dt is x². This is why the correct answer is that the derivative is equal to x². The other options do not represent the correct interpretation of the Fundamental Theorem or the nature of the function being considered.

To find the derivative of the integral ∫ from 0 to x of t² dt with respect to x, we can apply the Fundamental Theorem of Calculus. This theorem states that if F(x) is an antiderivative of f(t) on an interval [a, b], then the derivative of the integral from a to x of f(t) dt is simply f(x).

In this case, the function we are integrating is t². According to the Fundamental Theorem of Calculus, the derivative of the integral with respect to the upper limit (in this case x) is just the integrand evaluated at x. Therefore, we replace t with x in the function t²:

f(x) = x².

Thus, the derivative of the integral ∫ from 0 to x of t² dt is x². This is why the correct answer is that the derivative is equal to x².

The other options do not represent the correct interpretation of the Fundamental Theorem or the nature of the function being considered.