Question

If \( \int f(x) dx = F(x) + C \), then \( \frac{d}{dt} \int_{g(t)}^{h(t)} f(x)\,dx = \)

• A. \( f(h(t)) - f(g(t)) \)
• B. \( F(h(t)) - F(g(t)) \)
• C. \( F(h(t))f'(t) - F(g(t))g'(t) \)
• D. \( f(h(t))h'(t) - f(g(t))g'(t) \)

Hint:

Leibniz Rule for Variable Limits}
Upper limit → positive term with derivative
Lower limit → subtract the product
Works even when \( f(x) \) has no explicit \( t \)

Answer 1

The Correct Option is D

Use Leibniz Rule for derivative of definite integral with variable limits: \[ \frac{d}{dt} \left( \int_{g(t)}^{h(t)} f(x) dx \right) = f(h(t)) h'(t) - f(g(t)) g'(t) \] This handles both changing limits and fixed integrand.