Question Number 127349 by bemath last updated on 29/Dec/20
$$\:\int\:\frac{\left(\mathrm{1}−{s}^{\mathrm{2}} \right)^{\mathrm{5}/\mathrm{2}} }{{s}^{\mathrm{8}} }\:{ds}\:=? \\ $$
Commented by liberty last updated on 29/Dec/20
waw ... il-mistoqsija tiegħek hija interessanti ħafna
Answered by liberty last updated on 29/Dec/20
$${I}=\int\:\frac{{s}^{\mathrm{5}} \left({s}^{−\mathrm{2}} −\mathrm{1}\right)^{\mathrm{5}/\mathrm{2}} }{{s}^{\mathrm{8}} }\:{ds}\: \\ $$$${I}=\int\:{s}^{−\mathrm{3}} \left({s}^{−\mathrm{2}} −\mathrm{1}\right)^{\mathrm{5}/\mathrm{2}} \:{ds}\: \\ $$$$\:{let}\:\mu\:=\:{s}^{−\mathrm{2}} −\mathrm{1}\:\Rightarrow{d}\mu\:=\:−\mathrm{2}{s}^{−\mathrm{3}} \:{ds}\: \\ $$$${I}=−\frac{\mathrm{1}}{\mathrm{2}}\int\:\mu^{\mathrm{5}/\mathrm{2}} \:{d}\mu\:=\:−\frac{\mathrm{1}}{\mathrm{7}}\mu^{\mathrm{7}/\mathrm{2}} \:+\:{c} \\ $$$${I}=−\frac{\mathrm{1}}{\mathrm{7}}\left(\frac{\mathrm{1}−{s}^{\mathrm{2}} }{{s}^{\mathrm{2}} }\right)^{\mathrm{7}/\mathrm{2}} +{c}\:=\:−\frac{\left(\mathrm{1}−{s}^{\mathrm{2}} \right)^{\mathrm{7}/\mathrm{2}} }{{s}^{\mathrm{7}} }\:+\:{c} \\ $$