Sei integrali definiti immediati

Testo

Calcola i seguenti integrali definiti immediati: a) $\int_0^2(2x^3-x)\,dx$ b) $\int_0^{\ln 2}e^{2x}\,dx$ c) $\int_0^{\pi/3}\cos x\,dx$ d) $\int_1^2\dfrac{dx}{x^2}$ e) $\int_0^1 e^{-x}\,dx$ f) $\int_{-1}^1(4-x^2)\,dx$

Soluzione

a) $\left[\dfrac{x^4}{2}-\dfrac{x^2}{2}\right]_0^2=8-2=6.$

b) $\left[\dfrac{e^{2x}}{2}\right]_0^{\ln 2}=\dfrac{4-1}{2}=\dfrac32.$

c) $\big[\sin x\big]_0^{\pi/3}=\dfrac{\sqrt3}{2}\approx0.8660.$

d) $\left[-\dfrac1x\right]_1^2=-\dfrac12+1=\dfrac12.$

e) $\big[-e^{-x}\big]_0^1=1-\dfrac1e\approx0.6321.$

f) $\left[4x-\dfrac{x^3}{3}\right]_{-1}^1=\left(4-\dfrac13\right)-\left(-4+\dfrac13\right)=\dfrac{22}{3}\approx7.333.$

$\boxed{a=6,\ b=\tfrac32,\ c=\tfrac{\sqrt3}{2},\ d=\tfrac12,\ e=1-\tfrac1e,\ f=\tfrac{22}{3}}$

Collegamenti

Argomenti: Integrale
Concetti: Integrale definito
Competenze: Integrare
Tipo di esercizio: Calcolo integrale