Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-014 · ★☆☆ Basic
If \(F(x)=\displaystyle\int_1^x\cos(t^2)\,dt\), what is \(F'(x)\)?
Choose one option.
Show answer
- D. \(\cos(x^2)\)
Explanation
The integrand \(\cos(t^2)\) is continuous, so \(F'(x)=\cos(x^2)\).
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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