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

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

 

Exercice 14-10

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

Solution

$[x\operatorname {e} ^{x}]_{0}^{1}-\int _{0}^{1}\operatorname {e} ^{x}\,\mathrm {d} x+[x^{3}/3]_{0}^{1}-[t\ln t-t]_{1}^{2}=\mathrm {e} -(\mathrm {e} -1)+{\frac {1}{3}}-2\ln 2+1={\frac {7}{3}}-2\ln 2$.

 

Exercice 14-11

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

Solution

$\left[-{\frac {\ln(1+x)}{x}}\right]_{1}^{2}+\int _{1}^{2}{\frac {\mathrm {d} x}{x(1+x)}}=-{\frac {\ln 3}{2}}+\ln 2+\int _{1}^{2}\left({\frac {1}{x}}-{\frac {1}{x+1}}\right)\,\mathrm {d} x=-{\frac {\ln 3}{2}}+\ln 2+\left[\ln {\frac {x}{x+1}}\right]_{1}^{2}=-{\frac {3\ln 3}{2}}+3\ln 2$.