Exercices

Exercice 1   Écrire sans calcul les primitives suivantes.

$$\int_c^x \frac{\ln(t)}{t} \mathrm{d}t \quad;\quad \int_c^x t\ma... ... \mathrm{d}t \quad;\quad \int_c^x\mathrm{e}^{t+\mathrm{e}^t} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{2+\mathrm{e}^{2t}} \mathrm{d}t \quad;\quad \i... ...m{d}t \quad;\quad \int_c^x \frac{\arcsin^2(t)}{\sqrt{1-t^2}} \mathrm{d}t\;;$$

$$\int_c^x \frac{\sin(t)}{2+\cos(t)} \mathrm{d}t \quad;\quad \in... ...{d}t \quad;\quad \int_c^x \frac{\cos(t)}{\sqrt{1-\sin^2(t)}} \mathrm{d}t\;;$$

$$\int_c^x \tan^3(t)+\tan^5(t) \mathrm{d}t \quad;\quad \int_c^x ... ...thrm{d}t \quad;\quad \int_c^x \frac{1}{\cos^2(t)(3+\tan(t))} \mathrm{d}t\;;$$

$$\int_c^x \frac{2t+3}{(t^2+3t+5)^2} \mathrm{d}t \quad;\quad \int_... ... \mathrm{d}t \quad;\quad \int_c^x \frac{t+1}{\sqrt{t^2+2t+3}} \mathrm{d}t\;;$$

$$\int_c^x \mathrm{e}^{\sqrt{\sin(t)}}\frac{\cos(t)}{\sqrt{\sin(t)}... ...+\sqrt{\cos(\mathrm{e}^{-t})}}} {\sqrt{\cos(\mathrm{e}^{-t})}} \mathrm{d}t\;;$$

$$\int_c^x \frac{\mathrm{e}^{2t}(\sin(\mathrm{e}^{2t})-\sin^3(\math... ...s^3(\mathrm{e}^{2t})}\right)\sqrt{1+\cos^3(\mathrm{e}^{2t})}} \mathrm{d}t\;.$$

Exercice 2   Calculer les primitives suivantes, en utilisant les propriétés de l'exponentielle.

$$\int_c^x \cos^3(t) \mathrm{d}t \quad;\quad \int_c^x \sin^3(t) ... ... \int_c^x\cos^4(t) \mathrm{d}t \quad;\quad \int_c^x\sin^4(t) \mathrm{d}t\;;$$

$$\int_c^x\cos^2(t)\sin^2(t) \mathrm{d}t \quad;\quad \int_c^x\cos(t)\sin^3(t) \mathrm{d}t \quad;\quad \int_c^x\cos^3(t)\sin(t) \mathrm{d}t\;;$$

$$\int_c^x\cos^3(t)\sin^2(t) \mathrm{d}t \quad;\quad \int_c^x\cos^2(t)\sin^3(t) \mathrm{d}t \quad;\quad \int_c^x\cos(t)\sin^4(t) \mathrm{d}t\;;$$

$$\int_c^x \cosh^3(t) \mathrm{d}t \quad;\quad \int_c^x \sinh^3(t)... ...int_c^x\cosh^4(t) \mathrm{d}t \quad;\quad \int_c^x\sinh^4(t) \mathrm{d}t\;;$$

$$\int_c^x\cosh^2(t)\sinh^2(t) \mathrm{d}t \quad;\quad \int_c^x\c... ...sinh^3(t) \mathrm{d}t \quad;\quad \int_c^x\cosh^3(t)\sinh(t) \mathrm{d}t\;;$$

$$\int_c^x\cosh^3(t)\sinh^2(t) \mathrm{d}t \quad;\quad \int_c^x\c... ...sinh^3(t) \mathrm{d}t \quad;\quad \int_c^x\cosh(t)\sinh^4(t) \mathrm{d}t\;;$$

$$\int_c^x \mathrm{e}^t\cos(t) \mathrm{d}t \quad;\quad \int_c^x \m... ...(2t) \mathrm{d}t \quad;\quad \int_c^x \mathrm{e}^{-t}\sin(2t) \mathrm{d}t\;;$$

$$\int_c^x \mathrm{e}^{2t}\cos(t-\frac{\pi}{4}) \mathrm{d}t \quad;\quad \int_c^x \mathrm{e}^{-2t}\cos(t+\frac{\pi}{4}) \mathrm{d}t\;;$$

$$\int_c^x \mathrm{e}^{t}\cos(2t-\frac{\pi}{3}) \mathrm{d}t \quad;\quad \int_c^x \mathrm{e}^{-t}\cos(2t+\frac{\pi}{3}) \mathrm{d}t\;;$$

Exercice 3   Calculer les primitives suivantes, en utilisant l'intégration par parties.

$$\int_c^x t\mathrm{e}^t \mathrm{d}t \quad;\quad \int_c^x t^2\mathrm{e}^t \mathrm{d}t \quad;\quad \int_c^x t^3\mathrm{e}^t \mathrm{d}t\;;$$

$$\int_c^x t\ln(t) \mathrm{d}t \quad;\quad \int_c^x t^2\ln(t) \mathrm{d}t \quad;\quad \int_c^x t^3\ln(t) \mathrm{d}t\;;$$

$$\int_c^x t\sin(t) \mathrm{d}t \quad;\quad \int_c^x t^2\sin(t) \mathrm{d}t \quad;\quad \int_c^x t^3\sin(t) \mathrm{d}t\;;$$

$$\int_c^x t\cos(t) \mathrm{d}t \quad;\quad \int_c^x t^2\cos(t) \mathrm{d}t \quad;\quad \int_c^x t^3\cos(t) \mathrm{d}t\;;$$

$$\int_c^x \arcsin(t) \mathrm{d}t \quad;\quad \int_c^x t\arcsin(t) \mathrm{d}t \quad;\quad \int_c^x \arctan(t) \mathrm{d}t\;;$$

$$\int_c^x (t^2+1)\arctan(t) \mathrm{d}t \quad;\quad \int_c^x (t... ...thrm{e}^{3t} \mathrm{d}t \quad;\quad \int_c^x (t+1)\arcsin(t) \mathrm{d}t\;.$$

Exercice 4   Calculer les primitives de fractions rationnelles suivantes.

$$\int_c^x \frac{1}{t(t-1)} \mathrm{d}t \quad;\quad \int_c^x \frac{1}{t^2-1} \mathrm{d}t \quad;\quad \int_c^x \frac{1}{t(t^2-1)} \mathrm{d}t\;;$$

$$\int_c^x \frac{t}{t^2+4}\,\mathrm{d}t \quad;\quad \int_c^x \frac{t^3}{t^2+4}\,\mathrm{d}t \quad;\quad \int_c^x \frac{t^5}{t^2+3}\,\mathrm{d}t\;;$$

$$\int_c^x \frac{1}{t^2+4}\,\mathrm{d}t \quad;\quad \int_c^x \frac{... ...4}\,\mathrm{d}t \quad;\quad \int_c^x \frac{3t^2+2}{(t^2+4)(t-1)}\,\mathrm{d}t;$$

$$\int_c^x \frac{1}{t^2(t^2-1)} \mathrm{d}t \quad;\quad \int_c^x \... ...1)^2} \mathrm{d}t \quad;\quad \int_c^x \frac{1}{t^2(t^2-1)^2} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{(t^2-1)^2} \mathrm{d}t \quad;\quad \int_c^x \f... ...2+1)^2} \mathrm{d}t \quad;\quad \int_c^x \frac{1}{t(t^2+1)^2} \mathrm{d}t\;;$$

$$\int_c^x \frac{2t+3}{(t-2)(t+5)} \mathrm{d}t \quad;\quad \int_c^... ...1)(t+3)} \mathrm{d}t \quad;\quad \int_c^x \frac{1}{t^4-t^2-2} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{(t+2)(t^2+2t+5)} \mathrm{d}t \quad;\quad \int_... ...2)^3} \mathrm{d}t \quad;\quad \int_c^x \frac{t^4+1}{t(t-1)^3} \mathrm{d}t\;.$$

Exercice 5   Calculer les primitives suivantes.

$$\int_c^x \frac{1}{1+\mathrm{e}^t} \mathrm{d}t \quad;\quad \int_c... ...}t \quad;\quad \int_c^x \frac{\mathrm{e}^{2t}}{1-\mathrm{e}^t} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{\mathrm{e}^{2t}+\mathrm{e}^t+1} \mathrm{d}t \q... ..._c^x \frac{\mathrm{e}^{2t}+1}{2\mathrm{e}^t+\mathrm{e}^{-t}+1} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{1+\cosh(t)} \mathrm{d}t \quad;\quad \int_c^x \... ...rm{d}t \quad;\quad \int_c^x \frac{\cosh(t)}{\sinh(t)+\cosh(t)} \mathrm{d}t\;.$$

Exercice 6   Calculer les primitives suivantes.

$$\int_c^x \frac{1}{\sin(t)} \mathrm{d}t \quad;\quad \int_c^x \fra... ...t)} \mathrm{d}t \quad;\quad \int_c^x \frac{1}{\sin(t)\cos(t)} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{1+\sin(t)} \mathrm{d}t \quad;\quad \int_c^x\fr... ...{d}t \quad;\quad \int_c^x \frac{2+3\sin(t)}{5-4\sin(t)}\cos(t) \mathrm{d}t\;;$$

$$\int_c^x \frac{\tan(t) +2}{\tan(t) -1} \mathrm{d}t \quad;\quad \... ...cos(t)} \mathrm{d}t \quad;\quad \int_c^x \frac{1}{2+\sin(t)} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{\sin(t)(1+\cos^2(t))} \mathrm{d}t \quad;\quad ... ...d}t \quad;\quad \int_c^x \frac{\cos(t)}{\sin^2(t)+2\tan^2(t)} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{\sin(t)+\sin(2t)} \mathrm{d}t \quad;\quad \int... ...hrm{d}t \quad;\quad \int_c^x \frac{\cos(t)}{\sin(2t)\cos(3t)} \mathrm{d}t\;.$$

Exercice 7   Calculer les primitives suivantes.

$$\int_c^x \frac{1}{t+\sqrt{t-1}} \mathrm{d}t \quad;\quad \int_c^x... ...1-t}} \mathrm{d}t \quad\quad \int_c^x t\sqrt{\frac{t-2}{t+1}} \mathrm{d}t\;.$$

$$\int_c^x \sqrt{\frac{t-1}{t}} \mathrm{d}t \quad;\quad \int_c^x \... ...;\quad \int_c^x \frac{\sqrt{4t^2-1}}{\sqrt{2t+1}+2\sqrt{2t-1}} \mathrm{d}t\;;$$

Exercice 8   Calculer les primitives suivantes.

$$\int_c^x \frac{1}{\sqrt{t^2+2t}} \mathrm{d}t \quad;\quad \int_c^... ...}} \mathrm{d}t \quad;\quad \int_c^x \frac{t^2}{\sqrt{t^2+2t}} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{t+\sqrt{t(4-t)}} \mathrm{d}t \quad;\quad \int_... ...t)}} \mathrm{d}t \quad;\quad \int_c^x\frac{1}{t+\sqrt{t^2+1}} \mathrm{d}t\;;$$

$$\int_c^x \frac{t}{1+\sqrt{t(t-1)}} \mathrm{d}t \quad;\quad \int_... ...{2t^2+8t+1}} \mathrm{d}t \quad;\quad \int_c^x t\sqrt{-2t^2+t} \mathrm{d}t\;;$$

$$\int_c^x \frac{1}{\sqrt{-12t^2-12t-2}} \mathrm{d}t \quad;\quad \... ...}} \mathrm{d}t \quad;\quad \int_c^x \frac{t^2}{\sqrt{t^2-2t}} \mathrm{d}t\;.$$