Question Number 209557 by Spillover last updated on 14/Jul/24
$${Given}\: \\ $$$$\:\int_{\mathrm{2}} ^{\mathrm{4}} \left({ax}^{{n}} +\mathrm{1}\right){dx}=\mathrm{58}\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \left({ax}^{{n}} +\mathrm{1}\right){dx}=\mathrm{10} \\ $$$${find}\:{the}\:{value}\:{of}\:\:{a}\:\:\:{and}\:\:{n} \\ $$
Answered by mr W last updated on 14/Jul/24
$$\int_{\mathrm{0}} ^{\mathrm{2}} \left({ax}^{{n}} +\mathrm{1}\right){dx}=\left[\frac{{ax}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}+{x}\right]_{\mathrm{0}} ^{\mathrm{2}} =\frac{{a}\mathrm{2}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}+\mathrm{2}=\mathrm{10} \\ $$$$\Rightarrow\frac{{a}\mathrm{2}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}=\mathrm{8}\:\:\:…\left({i}\right) \\ $$$$\int_{\mathrm{0}} ^{\mathrm{4}} \left({ax}^{{n}} +\mathrm{1}\right){dx}=\frac{{a}\mathrm{4}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}+\mathrm{4}=\mathrm{10}+\mathrm{58}=\mathrm{68} \\ $$$$\Rightarrow\frac{{a}\mathrm{4}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}=\mathrm{64}\:\:\:…\left({ii}\right) \\ $$$$\left({ii}\right)/\left({i}\right):\:\left(\frac{\mathrm{4}}{\mathrm{2}}\right)^{{n}+\mathrm{1}} =\mathrm{8}\:\Rightarrow{n}=\mathrm{2}\:\checkmark \\ $$$$\Rightarrow{a}=\frac{\mathrm{8}×\mathrm{3}}{\mathrm{2}^{\mathrm{3}} }=\mathrm{3}\:\checkmark \\ $$
Commented by Spillover last updated on 14/Jul/24
$${thank}\:{you} \\ $$
Answered by lepuissantcedricjunior last updated on 16/Jul/24
$$\int_{\mathrm{2}} ^{\mathrm{4}} \left(\boldsymbol{{ax}}^{\boldsymbol{{n}}} +\mathrm{1}\right)\boldsymbol{{dx}}=\mathrm{58} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}} \left(\boldsymbol{{ax}}^{\boldsymbol{{n}}} +\mathrm{1}\right)\boldsymbol{{dx}}=\mathrm{10} \\ $$$$=>\begin{cases}{\int_{\mathrm{2}} ^{\mathrm{4}} \left(\boldsymbol{{ax}}^{\boldsymbol{{n}}} +\mathrm{1}\right)\boldsymbol{{dx}}}\\{\int_{\mathrm{0}} ^{\mathrm{2}} \left(\boldsymbol{{ax}}^{\boldsymbol{{n}}} +\mathrm{1}\right)\boldsymbol{{dx}}}\end{cases}=>\begin{cases}{=\left[\frac{\boldsymbol{{a}}}{\mathrm{1}+\boldsymbol{{n}}}\boldsymbol{{x}}^{\boldsymbol{{n}}+\mathrm{1}} +\boldsymbol{{x}}\right]_{\mathrm{2}} ^{\mathrm{4}} }\\{=\left[\frac{\boldsymbol{{a}}}{\boldsymbol{{n}}+\mathrm{1}}\boldsymbol{{x}}^{\boldsymbol{{n}}+\mathrm{1}} +\boldsymbol{{x}}\right]_{\mathrm{0}} ^{\mathrm{2}} }\end{cases}=>\begin{cases}{=\frac{\mathrm{2}^{\mathrm{2}\boldsymbol{{n}}+\mathrm{2}} \boldsymbol{{a}}}{\mathrm{1}+\boldsymbol{{n}}}−\frac{\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} \boldsymbol{{a}}}{\boldsymbol{{n}}+\mathrm{1}}+\mathrm{2}}\\{=\frac{\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} \boldsymbol{{a}}}{\boldsymbol{{n}}+\mathrm{1}}+\mathrm{2}}\end{cases} \\ $$$$=>\begin{cases}{\frac{\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} \boldsymbol{{a}}\left(\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} −\mathrm{1}\right)}{\boldsymbol{{n}}+\mathrm{1}}=\mathrm{56}\left(\mathrm{1}\right)}\\{\frac{\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} \boldsymbol{{a}}}{\boldsymbol{{n}}+\mathrm{1}}=\mathrm{8}\left(\mathrm{2}\right)\boldsymbol{{remplacons}}\:\boldsymbol{{dans}}\:\left(\mathrm{1}\right)}\end{cases} \\ $$$$=>\mathrm{8}\left(\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} −\mathrm{1}\right)=\mathrm{56}=>\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} =\mathrm{8}=\mathrm{2}^{\mathrm{3}} \\ $$$$=>\boldsymbol{{n}}+\mathrm{1}=\mathrm{3}=>\boldsymbol{{n}}=\mathrm{2} \\ $$$$\boldsymbol{{deplus}}\:\frac{\mathrm{2}^{\boldsymbol{{n}}+\mathrm{1}} \boldsymbol{{a}}}{\boldsymbol{{n}}+\mathrm{1}}=\mathrm{8}=>\frac{\mathrm{8}}{\mathrm{3}}\boldsymbol{{a}}=\mathrm{8}=>\boldsymbol{{a}}=\mathrm{3} \\ $$$$\boldsymbol{{d}}'\boldsymbol{{ou}}'\:\Rightarrow\left\{\left(\boldsymbol{{n}};\boldsymbol{{a}}\right)\right\}=\left\{\left(\mathrm{2};\mathrm{3}\right)\right\} \\ $$$$…………{le}\:{puissant}\:{cedric}\:{junior}…….. \\ $$
Commented by Spillover last updated on 30/Jul/24
$${great} \\ $$