Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-009 · ★★★ Advanced
By the fundamental theorem, what is \(\dfrac{d}{dx}\displaystyle\int_{0}^{x^{2}} \sin(t)\,dt\)?
Choose one option.
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- D. \(2x\sin(x^{2})\)
Explanation
The upper limit is \(x^{2}\), so evaluate the integrand at \(x^{2}\) and multiply by the derivative of \(x^{2}\): \(\sin(x^{2})\cdot 2x\).
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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