Intégration en mathématiques/Exercices/Intégrales 6

Calculer les intégrales suivantes.

Exercice 14-1

$\int _{0}^{1}{\frac {x}{3}}\operatorname {e} ^{x^{2}}\,\mathrm {d} x$

Solution

$\left[{\frac {\mathrm {e} ^{x^{2}}}{6}}\right]_{0}^{1}={\frac {\mathrm {e} -1}{6}}$ .

Exercice 14-2

$\int _{0}^{\frac {\pi }{2}}5\sin t\operatorname {e} ^{\cos t}\,\mathrm {d} t$

Solution

$\left[-5\operatorname {e} ^{\cos t}\right]_{0}^{\pi /2}=5(\mathrm {e} -1)$ .

Exercice 14-3

$\int _{-1}^{0}\left(x^{2}+x\right)\operatorname {e} ^{2x}\,\mathrm {d} x$

Solution

$\left[{\frac {x^{2}\operatorname {e} ^{2x}}{2}}\right]_{-1}^{0}=-{\frac {1}{2\mathrm {e} ^{2}}}$ .

Exercice 14-4

$\int _{\frac {1}{3}}^{1}\left(3t^{2}-4t+1\right)\operatorname {e} ^{t}\,\mathrm {d} t$

Solution

$\left[\left(\left(3t^{2}-4t+1\right)-\left(6t-4\right)+6\right)\operatorname {e} ^{t}\right]_{1/3}^{1}=\left[\left(3t^{2}-10t+11\right)\operatorname {e} ^{t}\right]_{1/3}^{1}=4\operatorname {e} -8{\sqrt[{3}]{\mathrm {e} }}$ .

Exercice 14-5

$\int _{1}^{2}{\frac {\cos(\ln t)}{t}}\,\mathrm {d} t$

Solution

$\int _{0}^{\ln 2}\cos =\left[\sin \right]_{0}^{\ln 2}=\sin \left(\ln 2\right)$ .

Exercice 14-6

$\int _{\pi /3}^{\pi /2}\tan(x/2)\ln(1+\cos x)\,\mathrm {d} x$

Solution

$\left[{\frac {-\ln ^{2}(1+\cos )}{2}}\right]_{\pi /3}^{\pi /2}={\frac {\ln ^{2}(3/2)}{2}}$ .

Exercice 14-7

$\int _{0}^{1}x^{2}\operatorname {e} ^{-x^{3}}\,\mathrm {d} x$

Solution

$\left[{\frac {-\mathrm {e} ^{-x^{3}}}{3}}\right]_{0}^{1}={\frac {1-\mathrm {e} ^{-1}}{3}}$ .

Exercice 14-8

$\int _{0}^{\mathrm {e} -1}\ln(1+x)\,\mathrm {d} x$

Solution

$\left[t\ln t-t\right]_{1}^{\mathrm {e} }=1$ .

Exercice 14-9

$\int _{1}^{\mathrm {e} }{\frac {\mathrm {d} t}{t{\sqrt {1-(\ln t)^{2}}}}}$

Solution

$\int _{0}^{1}{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}=\int _{0}^{\pi /2}{\frac {\cos t\,\mathrm {d} t}{\sqrt {1-\sin ^{2}t}}}=\int _{0}^{\pi /2}\mathrm {d} t={\frac {\pi }{2}}$ .

Cet article est issu de Wikiversity. Le texte est sous licence Creative Commons - Attribution - Partage dans les Mêmes. Des conditions supplémentaires peuvent s'appliquer aux fichiers multimédias.