Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-011 · ★☆☆ Basic
Compute \(\displaystyle\frac{d}{dx}\int_0^x t^2\,dt\).
Choose one option.
Show answer
- A. \(x^2\)
Explanation
By the fundamental theorem, the derivative of \(\int_0^x f\) is \(f(x)\), here \(x^2\). Integrating \(t^2\) without differentiating would give \(x^3/3\).
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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