Exercice 2 : Primitives "décalées", à reconnaître a) $$ \int\left(x^{2}-3 x+4\right)^{5}(4 x-6) d x $$ b) $$ \int \sin ^{7} x \cos x d x $$ c) $$ \int \frac{\sin x}{\cos x} d x $$ d) $$ \int \frac{9 x^{2}-6}{x^{3}-2 x+5} d x $$ e) $$ \int \frac{3 x}{\sqrt{x^{2}-2}} d x $$ f) $$ \int e^{2 x^{2}-4 x}(x-1) d x $$ g) $$ \int \frac{3}{1-\sin ^{2} x} d x $$ h) $$ \int \frac{2 x}{\sqrt{2-x^{2}}} \mathrm{~d} x $$ i) $$ \int_{-1}^{1} \frac{x}{\left(x^{2}-4\right)^{4}} d x $$

Question

UpStudy AI · Accepted Answer

Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int_{-1}^{1} \frac{x}{\left(x^{2}-4\right)^{4}} dx$$
- step1: Evaluate the integral:
  $$\int \frac{x}{\left(x^{2}-4\right)^{4}} dx$$
- step2: Use the substitution $$dx=\frac{1}{2x} dt$$ to transform the integral$$:$$
  $$\int \frac{x}{\left(x^{2}-4\right)^{4}}\times \frac{1}{2x} dt$$
- step3: Simplify:
  $$\int \frac{1}{2\left(x^{2}-4\right)^{4}} dt$$
- step4: Use the substitution $$t=x^{2}$$ to transform the integral$$:$$
  $$\int \frac{1}{2\left(t-4\right)^{4}} dt$$
- step5: Rewrite the expression:
  $$\int \frac{1}{2}\times \frac{1}{\left(t-4\right)^{4}} dt$$
- step6: Use properties of integrals:
  $$\frac{1}{2}\times \int \frac{1}{\left(t-4\right)^{4}} dt$$
- step7: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\frac{1}{2}\times \int \frac{1}{\left(t-4\right)^{4}}\times 1 dv$$
- step8: Simplify:
  $$\frac{1}{2}\times \int \frac{1}{\left(t-4\right)^{4}} dv$$
- step9: Use the substitution $$v=t-4$$ to transform the integral$$:$$
  $$\frac{1}{2}\times \int \frac{1}{v^{4}} dv$$
- step10: Evaluate the integral:
  $$\frac{1}{2}\times \frac{v^{-4+1}}{-4+1}$$
- step11: Simplify:
  $$\frac{1}{2}\times \frac{v^{-3}}{-3}$$
- step12: Rewrite the expression:
  $$\frac{1}{2}\left(-\frac{v^{-3}}{3}\right)$$
- step13: Multiply the terms:
  $$-\frac{1}{2}\times \frac{v^{-3}}{3}$$
- step14: Multiply the terms:
  $$-\frac{v^{-3}}{2\times 3}$$
- step15: Multiply the terms:
  $$-\frac{v^{-3}}{6}$$
- step16: Simplify:
  $$-\frac{1}{6v^{3}}$$
- step17: Substitute back:
  $$-\frac{1}{6\left(t-4\right)^{3}}$$
- step18: Substitute back:
  $$-\frac{1}{6\left(x^{2}-4\right)^{3}}$$
- step19: Return the limits:
  $$\left(-\frac{1}{6\left(x^{2}-4\right)^{3}}\right)\bigg |_{-1}^{1}$$
- step20: Calculate the value:
  $$0$$
Calculate the integral $$ \int \frac{9 x^{2}-6}{x^{3}-2 x+5} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{9x^{2}-6}{x^{3}-2x+5} dx$$
- step1: Use the substitution $$dx=\frac{1}{3x^{2}-2} dt$$ to transform the integral$$:$$
  $$\int \frac{9x^{2}-6}{x^{3}-2x+5}\times \frac{1}{3x^{2}-2} dt$$
- step2: Simplify:
  $$\int \frac{3}{x^{3}-2x+5} dt$$
- step3: Use the substitution $$t=x^{3}-2x+5$$ to transform the integral$$:$$
  $$\int \frac{3}{t} dt$$
- step4: Rewrite the expression:
  $$\int 3\times \frac{1}{t} dt$$
- step5: Use properties of integrals:
  $$3\times \int \frac{1}{t} dt$$
- step6: Evaluate the integral:
  $$3\ln{\left(\left|t\right|\right)}$$
- step7: Substitute back:
  $$3\ln{\left(\left|x^{3}-2x+5\right|\right)}$$
- step8: Add the constant of integral C:
  $$3\ln{\left(\left|x^{3}-2x+5\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \sin ^{7} x \cos x d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \sin^{7}\left(x\right)\cos\left(x\right) dx$$
- step1: Evaluate the integral:
  $$\frac{\sin^{7+1}\left(x\right)}{7+1}$$
- step2: Add the numbers:
  $$\frac{\sin^{8}\left(x\right)}{7+1}$$
- step3: Add the numbers:
  $$\frac{\sin^{8}\left(x\right)}{8}$$
- step4: Simplify the expression:
  $$\frac{1}{8}\sin^{8}\left(x\right)$$
- step5: Add the constant of integral C:
  $$\frac{1}{8}\sin^{8}\left(x\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int e^{2 x^{2}-4 x}(x-1) d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int e^{2x^{2}-4x}\left(x-1\right) dx$$
- step1: Use the substitution $$dx=\frac{1}{4x-4} dt$$ to transform the integral$$:$$
  $$\int e^{2x^{2}-4x}\left(x-1\right)\times \frac{1}{4x-4} dt$$
- step2: Simplify:
  $$\int \frac{e^{2x^{2}-4x}}{4} dt$$
- step3: Use the substitution $$t=2x^{2}-4x$$ to transform the integral$$:$$
  $$\int \frac{e^{t}}{4} dt$$
- step4: Rewrite the expression:
  $$\int \frac{1}{4}e^{t} dt$$
- step5: Use properties of integrals:
  $$\frac{1}{4}\times \int e^{t} dt$$
- step6: Evaluate the integral:
  $$\frac{1}{4}e^{t}$$
- step7: Multiply the terms:
  $$\frac{e^{t}}{4}$$
- step8: Substitute back:
  $$\frac{e^{2x^{2}-4x}}{4}$$
- step9: Add the constant of integral C:
  $$\frac{e^{2x^{2}-4x}}{4} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{3 x}{\sqrt{x^{2}-2}} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{3x}{\sqrt{x^{2}-2}} dx$$
- step1: Rewrite the expression:
  $$\int 3\times \frac{x}{\sqrt{x^{2}-2}} dx$$
- step2: Use properties of integrals:
  $$3\times \int \frac{x}{\sqrt{x^{2}-2}} dx$$
- step3: Rewrite the expression:
  $$3\times \int \frac{x}{\left(x^{2}-2\right)^{\frac{1}{2}}} dx$$
- step4: Use the substitution $$dx=\frac{1}{2x} dt$$ to transform the integral$$:$$
  $$3\times \int \frac{x}{\left(x^{2}-2\right)^{\frac{1}{2}}}\times \frac{1}{2x} dt$$
- step5: Simplify:
  $$3\times \int \frac{1}{2\left(x^{2}-2\right)^{\frac{1}{2}}} dt$$
- step6: Use the substitution $$t=x^{2}$$ to transform the integral$$:$$
  $$3\times \int \frac{1}{2\left(t-2\right)^{\frac{1}{2}}} dt$$
- step7: Rewrite the expression:
  $$3\times \int \frac{1}{2}\times \frac{1}{\left(t-2\right)^{\frac{1}{2}}} dt$$
- step8: Use properties of integrals:
  $$3\times \frac{1}{2}\times \int \frac{1}{\left(t-2\right)^{\frac{1}{2}}} dt$$
- step9: Multiply:
  $$\frac{3}{2}\times \int \frac{1}{\left(t-2\right)^{\frac{1}{2}}} dt$$
- step10: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\frac{3}{2}\times \int \frac{1}{\left(t-2\right)^{\frac{1}{2}}}\times 1 dv$$
- step11: Simplify:
  $$\frac{3}{2}\times \int \frac{1}{\left(t-2\right)^{\frac{1}{2}}} dv$$
- step12: Use the substitution $$v=t-2$$ to transform the integral$$:$$
  $$\frac{3}{2}\times \int \frac{1}{v^{\frac{1}{2}}} dv$$
- step13: Evaluate the integral:
  $$\frac{3}{2}\times \frac{v^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}$$
- step14: Simplify:
  $$\frac{3}{2}\times \frac{v^{\frac{1}{2}}}{\frac{1}{2}}$$
- step15: Multiply the terms:
  $$3v^{\frac{1}{2}}$$
- step16: Substitute back:
  $$3\left(t-2\right)^{\frac{1}{2}}$$
- step17: Substitute back:
  $$3\left(x^{2}-2\right)^{\frac{1}{2}}$$
- step18: Simplify:
  $$3\sqrt{x^{2}-2}$$
- step19: Add the constant of integral C:
  $$3\sqrt{x^{2}-2} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int\left(x^{2}-3 x+4\right)^{5}(4 x-6) d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \left(x^{2}-3x+4\right)^{5}\left(4x-6\right) dx$$
- step1: Multiply the terms:
  $$\int 2\left(x^{2}-3x+4\right)^{5}\left(2x-3\right) dx$$
- step2: Use properties of integrals:
  $$2\times \int \left(x^{2}-3x+4\right)^{5}\left(2x-3\right) dx$$
- step3: Use the substitution $$dx=\frac{1}{2x-3} dt$$ to transform the integral$$:$$
  $$2\times \int \left(x^{2}-3x+4\right)^{5}\left(2x-3\right)\times \frac{1}{2x-3} dt$$
- step4: Simplify:
  $$2\times \int \left(x^{2}-3x+4\right)^{5} dt$$
- step5: Use the substitution $$t=x^{2}-3x+4$$ to transform the integral$$:$$
  $$2\times \int t^{5} dt$$
- step6: Evaluate the integral:
  $$2\times \frac{t^{5+1}}{5+1}$$
- step7: Simplify:
  $$2\times \frac{t^{6}}{6}$$
- step8: Reduce the fraction:
  $$1\times \frac{t^{6}}{3}$$
- step9: Multiply the terms:
  $$\frac{t^{6}}{3}$$
- step10: Substitute back:
  $$\frac{\left(x^{2}-3x+4\right)^{6}}{3}$$
- step11: Add the constant of integral C:
  $$\frac{\left(x^{2}-3x+4\right)^{6}}{3} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{3}{1-\sin ^{2} x} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{3}{1-\sin^{2}\left(x\right)} dx$$
- step1: Rewrite the expression:
  $$\int 3\cos^{-2}\left(x\right) dx$$
- step2: Rewrite the expression:
  $$\int \frac{3}{\cos^{2}\left(x\right)} dx$$
- step3: Rewrite the expression:
  $$\int 3\times \frac{1}{\cos^{2}\left(x\right)} dx$$
- step4: Use properties of integrals:
  $$3\times \int \frac{1}{\cos^{2}\left(x\right)} dx$$
- step5: Evaluate the integral:
  $$3\tan\left(x\right)$$
- step6: Add the constant of integral C:
  $$3\tan\left(x\right) + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{\sin x}{\cos x} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \frac{\sin\left(x\right)}{\cos\left(x\right)} dx$$
- step1: Evaluate the integral:
  $$\ln{\left(\left|\sec\left(x\right)\right|\right)}$$
- step2: Add the constant of integral C:
  $$\ln{\left(\left|\sec\left(x\right)\right|\right)} + C, C \in \mathbb{R}$$
Calculate the integral $$ \int \frac{2 x}{\sqrt{2-x^{2}}} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{2x}{\sqrt{2-x^{2}}} dx$$
- step1: Rewrite the expression:
  $$\int 2\times \frac{x}{\sqrt{2-x^{2}}} dx$$
- step2: Use properties of integrals:
  $$2\times \int \frac{x}{\sqrt{2-x^{2}}} dx$$
- step3: Rewrite the expression:
  $$2\times \int \frac{x}{\left(2-x^{2}\right)^{\frac{1}{2}}} dx$$
- step4: Use the substitution $$dx=-\frac{1}{2x} dt$$ to transform the integral$$:$$
  $$2\times \int \frac{x}{\left(2-x^{2}\right)^{\frac{1}{2}}}\times \left(-\frac{1}{2x}\right) dt$$
- step5: Simplify:
  $$2\times \int -\frac{1}{2\left(2-x^{2}\right)^{\frac{1}{2}}} dt$$
- step6: Use the substitution $$t=-x^{2}$$ to transform the integral$$:$$
  $$2\times \int \frac{-1}{2\left(2+t\right)^{\frac{1}{2}}} dt$$
- step7: Rewrite the expression:
  $$2\times \int -\frac{1}{2}\times \frac{1}{\left(2+t\right)^{\frac{1}{2}}} dt$$
- step8: Use properties of integrals:
  $$2\left(-\frac{1}{2}\right)\times \int \frac{1}{\left(2+t\right)^{\frac{1}{2}}} dt$$
- step9: Multiply the numbers:
  $$-\int \frac{1}{\left(2+t\right)^{\frac{1}{2}}} dt$$
- step10: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$-\int \frac{1}{\left(2+t\right)^{\frac{1}{2}}}\times 1 dv$$
- step11: Simplify:
  $$-\int \frac{1}{\left(2+t\right)^{\frac{1}{2}}} dv$$
- step12: Use the substitution $$v=2+t$$ to transform the integral$$:$$
  $$-\int \frac{1}{v^{\frac{1}{2}}} dv$$
- step13: Evaluate the integral:
  $$-\frac{v^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}$$
- step14: Simplify:
  $$-\frac{v^{\frac{1}{2}}}{\frac{1}{2}}$$
- step15: Substitute back:
  $$-\frac{\left(2+t\right)^{\frac{1}{2}}}{\frac{1}{2}}$$
- step16: Substitute back:
  $$-\frac{\left(2-x^{2}\right)^{\frac{1}{2}}}{\frac{1}{2}}$$
- step17: Simplify:
  $$-\frac{\sqrt{2-x^{2}}}{\frac{1}{2}}$$
- step18: Add the constant of integral C:
  $$-\frac{\sqrt{2-x^{2}}}{\frac{1}{2}} + C, C \in \mathbb{R}$$
Voici les solutions pour chaque intégrale demandée :

a) Pour l'intégrale $$ \int\left(x^{2}-3 x+4\right)^{5}(4 x-6) d x $$ :
$$
\frac{(x^{2}-3x+4)^{6}}{3} + C, \quad C \in \mathbb{R}
$$

b) Pour l'intégrale $$ \int \sin ^{7} x \cos x d x $$ :
$$
\frac{1}{8}\sin^{8}(x) + C, \quad C \in \mathbb{R}
$$

c) Pour l'intégrale $$ \int \frac{\sin x}{\cos x} d x $$ :
$$
\ln{(\left|\sec(x)\right|)} + C, \quad C \in \mathbb{R}
$$

d) Pour l'intégrale $$ \int \frac{9 x^{2}-6}{x^{3}-2 x+5} d x $$ :
$$
3\ln{(\left|x^{3}-2x+5\right|)} + C, \quad C \in \mathbb{R}
$$

e) Pour l'intégrale $$ \int \frac{3 x}{\sqrt{x^{2}-2}} d x $$ :
$$
3\sqrt{x^{2}-2} + C, \quad C \in \mathbb{R}
$$

f) Pour l'intégrale $$ \int e^{2 x^{2}-4 x}(x-1) d x $$ :
$$
\frac{e^{2x^{2}-4x}}{4} + C, \quad C \in \mathbb{R}
$$

g) Pour l'intégrale $$ \int \frac{3}{1-\sin ^{2} x} d x $$ :
$$
3\tan(x) + C, \quad C \in \mathbb{R}
$$

h) Pour l'intégrale $$ \int \frac{2 x}{\sqrt{2-x^{2}}} d x $$ :
$$
-\frac{\sqrt{2-x^{2}}}{\frac{1}{2}} + C, \quad C \in \mathbb{R}
$$

i) Pour l'intégrale $$ \int_{-1}^{1} \frac{x}{\left(x^{2}-4\right)^{4}} d x $$ :
$$
0
$$

Ces résultats montrent les primitives des intégrales demandées, avec $$ C $$ représentant la constante d'intégration.
Voici les primitives des intégrales demandées : a) $$ \int\left(x^{2}-3 x+4\right)^{5}(4 x-6) d x = \frac{(x^{2}-3x+4)^{6}}{3} + C $$ b) $$ \int \sin ^{7} x \cos x d x = \frac{1}{8}\sin^{8}(x) + C $$ c) $$ \int \frac{\sin x}{\cos x} d x = \ln{(\left|\sec(x)\right|)} + C $$ d) $$ \int \frac{9 x^{2}-6}{x^{3}-2 x+5} d x = 3\ln{(\left|x^{3}-2x+5\right|)} + C $$ e) $$ \int \frac{3 x}{\sqrt{x^{2}-2}} d x = 3\sqrt{x^{2}-2} + C $$ f) $$ \int e^{2 x^{2}-4 x}(x-1) d x = \frac{e^{2x^{2}-4x}}{4} + C $$ g) $$ \int \frac{3}{1-\sin ^{2} x} d x = 3\tan(x) + C $$ h) $$ \int \frac{2 x}{\sqrt{2-x^{2}}} d x = -\sqrt{2-x^{2}} + C $$ i) $$ \int_{-1}^{1} \frac{x}{\left(x^{2}-4\right)^{4}} d x = 0 $$ Ces primitives sont obtenues en utilisant des méthodes de substitution et d'intégration par parties.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions