Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-001 · ★★☆ Intermediate
According to the fundamental theorem of calculus, \(\dfrac{d}{dx}\displaystyle\int_{1}^{x} \cos(t^{2})\,dt\) equals...
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- D. \(\cos(x^{2})\)
Explanation
The derivative of an integral with upper limit x returns the integrand evaluated at x: \(\cos(x^{2})\). There is no need to integrate first or use the chain rule, because the upper limit is exactly x.
Review the concept: Fundamental theorem of calculus
It links derivatives and integrals: a definite integral equals an antiderivative evaluated at the endpoints, and differentiating an integral with variable upper limit returns the integrand.
\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]
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