Integration Questions

Question Number 175343 by rexford last updated on 27/Aug/22

$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{5}} }{\:\sqrt{\mathrm{3}−{x}}}{dx} \\$$

Commented by BaliramKumar last updated on 28/Aug/22

$${I}\:{think}\:{Q}.\:{will}\:{be}\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\frac{\:{x}^{\mathrm{5}} }{\:\sqrt{\mathrm{3}−{x}}}{dx}\:=\:\frac{\mathrm{13824}\sqrt{\mathrm{3}}}{\mathrm{77}} \\$$

Answered by Ar Brandon last updated on 27/Aug/22

$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{5}} }{\:\sqrt{\mathrm{3}−{x}}}{dx}=\int_{\mathrm{0}} ^{\infty} \frac{\left(\mathrm{3}−\left(\mathrm{3}−{x}\right)\right)^{\mathrm{5}} }{\:\sqrt{\mathrm{3}−{x}}}{dx} \\$$$$=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{3}^{\mathrm{5}} −\mathrm{5}.\mathrm{3}^{\mathrm{4}} \sqrt{\mathrm{3}−{x}}+\mathrm{10}.\mathrm{3}^{\mathrm{3}} \sqrt{\left(\mathrm{3}−{x}\right)^{\mathrm{2}} }−\mathrm{10}.\mathrm{3}^{\mathrm{2}} \sqrt{\left(\mathrm{3}−{x}\right)^{\mathrm{3}} }+\mathrm{5}.\mathrm{3}\sqrt{\left(\mathrm{3}−{x}\right)^{\mathrm{4}} }−\sqrt{\left(\mathrm{3}−{x}\right)^{\mathrm{5}} }}{\:\sqrt{\mathrm{3}−{x}}}{dx} \\$$$$=\int_{\mathrm{0}} ^{\infty} \left(\mathrm{3}^{\mathrm{5}} \left(\mathrm{3}−{x}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} −\mathrm{5}.\mathrm{3}^{\mathrm{4}} +\mathrm{10}.\mathrm{3}^{\mathrm{3}} \sqrt{\mathrm{3}−{x}}−\mathrm{10}.\mathrm{3}^{\mathrm{2}} \left(\mathrm{3}−{x}\right)+\mathrm{5}.\mathrm{3}\left(\mathrm{3}−{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} −\left(\mathrm{3}−{x}\right)^{\mathrm{2}} \right){dx} \\$$$$\mathrm{1}.\mathrm{1} \\$$$$\mathrm{1}.\mathrm{2}.\mathrm{1} \\$$$$\mathrm{1}.\mathrm{3}.\mathrm{3}.\mathrm{1} \\$$$$\mathrm{1}.\mathrm{4}.\mathrm{6}.\mathrm{4}.\mathrm{1} \\$$$$\mathrm{1}.\mathrm{5}.\mathrm{10}.\mathrm{10}.\mathrm{5}.\mathrm{1} \\$$

Commented by Ar Brandon last updated on 27/Aug/22

$$\mathrm{Can}\:\mathrm{you}\:\mathrm{continue}? \\$$

Answered by CElcedricjunior last updated on 27/Aug/22

$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{x}}^{\mathrm{5}} }{\:\sqrt{\mathrm{3}−\boldsymbol{{x}}}}\boldsymbol{{dx}}=\boldsymbol{{k}} \\$$$$\boldsymbol{{posons}}\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}−\boldsymbol{{x}}}}=\boldsymbol{{a}}=>\boldsymbol{{x}}=\mathrm{3}−\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} } \\$$$$=>\boldsymbol{{dx}}=\frac{\mathrm{2}}{\boldsymbol{{a}}^{\mathrm{3}} }\boldsymbol{{da}} \\$$$$\boldsymbol{{qd}}:\begin{cases}{\boldsymbol{{x}}−>\mathrm{0}}\\{\boldsymbol{{x}}−>\infty}\end{cases}=>\begin{cases}{\boldsymbol{{a}}−>\mathrm{1}/\sqrt{\mathrm{3}}}\\{\boldsymbol{{a}}−>\mathrm{0}}\end{cases} \\$$$$\boldsymbol{{k}}=\mathrm{2}\int_{\mathrm{1}/\sqrt{\mathrm{3}}} ^{\mathrm{0}} \boldsymbol{{a}}\left(\mathrm{3}−\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} }\right)^{\mathrm{5}} \boldsymbol{{da}} \\$$$$\boldsymbol{{or}}\:\left(\mathrm{3}−\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} }\right)^{\mathrm{5}} =\underset{\boldsymbol{{p}}=\mathrm{0}} {\overset{\mathrm{5}} {\sum}}\boldsymbol{{C}}_{\mathrm{5}} ^{{p}} \mathrm{3}^{\mathrm{5}−\boldsymbol{{p}}} \left(−\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} }\right)^{\boldsymbol{{p}}} \\$$$$={C}_{\mathrm{5}} ^{\mathrm{0}} \mathrm{3}^{\mathrm{5}} −{C}_{\mathrm{5}} ^{\mathrm{1}} \frac{\mathrm{3}^{\mathrm{4}} }{{a}^{\mathrm{2}} }+{C}_{\mathrm{5}} ^{\mathrm{2}} \frac{\mathrm{3}^{\mathrm{3}} }{{a}^{\mathrm{4}} }−{C}_{\mathrm{5}} ^{\mathrm{3}} \frac{\mathrm{3}^{\mathrm{2}} }{\boldsymbol{{a}}^{\mathrm{6}} }+\boldsymbol{{C}}_{\mathrm{5}} ^{\mathrm{4}} \frac{\mathrm{3}}{\boldsymbol{{a}}^{\mathrm{8}} }−\boldsymbol{{C}}_{\mathrm{5}} ^{\mathrm{5}} \frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{10}} } \\$$$$=\mathrm{3}^{\mathrm{5}} −\frac{\mathrm{5}×\mathrm{81}}{\boldsymbol{{a}}^{\mathrm{2}} }+\frac{\mathrm{10}×\mathrm{27}}{\boldsymbol{{a}}^{\mathrm{4}} }−\frac{\mathrm{10}×\mathrm{9}}{\boldsymbol{{a}}^{\mathrm{6}} }+\frac{\mathrm{15}}{\boldsymbol{{a}}^{\mathrm{8}} }−\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{10}} } \\$$$$\boldsymbol{{k}}=−\infty \\$$

Answered by MJS_new last updated on 28/Aug/22

$$\int\frac{{x}^{\mathrm{5}} }{\:\sqrt{\mathrm{3}−{x}}}{dx}= \\$$$$\:\:\:\:\:\left[{t}=\mathrm{3}−{x}\:\rightarrow\:{dx}=−{dt}\right] \\$$$$=\int\frac{\left(\mathrm{3}−{t}\right)^{\mathrm{5}} }{\:\sqrt{{t}}}{dt}= \\$$$$=\int\left({t}^{\mathrm{9}/\mathrm{2}} −\mathrm{15}{t}^{\mathrm{7}/\mathrm{2}} +\mathrm{90}{t}^{\mathrm{5}/\mathrm{2}} −\mathrm{270}{t}^{\mathrm{3}/\mathrm{2}} +\mathrm{405}{t}^{\mathrm{1}/\mathrm{2}} −\mathrm{243}{t}^{−\mathrm{1}/\mathrm{2}} \right){dt}= \\$$$$=\frac{\mathrm{2}}{\mathrm{11}}{t}^{\mathrm{11}/\mathrm{2}} −\frac{\mathrm{10}}{\mathrm{3}}{t}^{\mathrm{9}/\mathrm{2}} +\frac{\mathrm{180}}{\mathrm{7}}{t}^{\mathrm{7}/\mathrm{2}} −\mathrm{108}{t}^{\mathrm{5}/\mathrm{2}} +\mathrm{270}{t}^{\mathrm{3}/\mathrm{2}} −\mathrm{486}{t}^{\mathrm{1}/\mathrm{2}} = \\$$$$...={F}\left({x}\right)+{C} \\$$$${F}\left({x}\right)=−\frac{\mathrm{2}\sqrt{\mathrm{3}−{x}}\left(\mathrm{21}{x}^{\mathrm{5}} +\mathrm{70}{x}^{\mathrm{4}} +\mathrm{240}{x}^{\mathrm{3}} +\mathrm{864}{x}^{\mathrm{2}} +\mathrm{3456}{x}+\mathrm{20736}\right)}{\mathrm{231}} \\$$$${F}\left(\mathrm{0}\right)=−\frac{\mathrm{13824}\sqrt{\mathrm{3}}}{\mathrm{77}} \\$$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{F}\left({x}\right)\:\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined} \\$$$$\Rightarrow\:\mathrm{integral}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist} \\$$