Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-005 · ★★☆ Intermediate
Compute \(\displaystyle\int_{0}^{1} e^{x}\,dx\).
Choose one option.
Show answer
- D. \(-1 + e\)
Explanation
An antiderivative is \(F(x)=e^{x}\). By the fundamental theorem, the integral is \(F(1)-F(0)=-1 + e\).
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-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-009 By the fundamental theorem, what is (d)/(dx)∫_0^x^2 sin (t) dt?
← ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-004 ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-006 →