Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-019 · ★★★ Advanced
If \(F(x)=\displaystyle\int_x^{x^2}t\,dt\), what is \(F'(x)\)?
Choose one option.
Show answer
- B. \(2x^3-x\)
Explanation
Differentiating each limit: \(F'(x)=x^2\cdot2x-x\cdot1=2x^3-x\). It agrees with \(F(x)=\tfrac{x^4-x^2}{2}\).
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.
\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]
More practice problems: fundamental theorem of calculus
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-001 According to the fundamental theorem of calculus, (d)/(dx)∫_1^x cos (t^2) dt equals...
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-002 Compute ∫_0^2 (3 x^2 + 1) dx.
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-003 Compute ∫_1^3 2 x dx.
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-004 Compute ∫_0^π sin x dx.
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-005 Compute ∫_0^1 e^x dx.
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-006 Compute ∫_1^e (1)/(x) dx.
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-007 Compute ∫_0^4 √x dx.
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-008 What does the second part of the fundamental theorem of calculus state?
← ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-018 ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-020 →