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) \( \int_{-1}^{1} \frac{x}{\left(x^{2}-4\right)^{4}} d x \)

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.