Exercice 1
Calculer \(\int_1^2\left(x + \frac{1}{x^2}\right)dx\), \(\int_0^{\ln2}e^{2x}\,dx\), \(\int_1^e\frac{1}{x}\,dx\).
Correction
\(\left[\frac{x^2}{2} - \frac1x\right]_1^2 = (2 - \frac12) - (\frac12 - 1) = 2\). \(\left[\frac{e^{2x}}{2}\right]_0^{\ln2} = 2 - \frac12 = \frac32\). \(1\).
Exercice 2
Calculer \(\int_0^1\frac{2x}{x^2 + 1}\,dx\).
Correction
\([\ln(x^2 + 1)]_0^1 = \ln2\).
Exercice 3
Calculer par parties \(\int_1^e\ln x\,dx\) et \(\int_0^1(x + 1)e^x\,dx\).
Correction
\([x\ln x - x]_1^e = 1\). \(u = x + 1\), \(v' = e^x\) : \([(x + 1)e^x]_0^1 - \int_0^1e^x\,dx = 2e - 1 - (e - 1) = e\).
Exercice 4
Calculer l'aire du domaine limité par \(y = e^x\), l'axe des abscisses et les droites \(x = 0\), \(x = 1\).
Correction
\(\int_0^1e^x\,dx = e - 1\) u.a.
Exercice 5
Calculer la valeur moyenne de \(f(x) = \sin x\) sur \([0 ; \pi]\).
Correction
\(\frac{1}{\pi}[-\cos x]_0^{\pi} = \frac{2}{\pi}\).