Integration Questions

Question Number 130486 by benjo_mathlover last updated on 26/Jan/21

$$\:\int_{\mathrm{0}} ^{\:{x}} \:\frac{\mathrm{cos}\:{t}\:\sqrt[{\mathrm{4}}]{\mathrm{sin}^{\mathrm{3}} \:{t}}}{\left(\mathrm{sin}\:{x}−\mathrm{sin}\:{t}\right)^{\mathrm{3}/\mathrm{4}} }\:{dt}\:? \\$$

Answered by MJS_new last updated on 26/Jan/21

$$\int\mathrm{cos}\:{t}\:\left(\frac{\mathrm{sin}\:{t}}{\mathrm{sin}\:{x}\:−\mathrm{sin}\:{t}}\right)^{\mathrm{3}/\mathrm{4}} {dt}= \\$$$$\:\:\:\:\:\left[{u}=\left(\frac{\mathrm{sin}\:{t}}{\mathrm{sin}\:{x}\:−\mathrm{sin}\:{t}}\right)^{\mathrm{1}/\mathrm{4}} \:\rightarrow\:{dt}=\frac{\mathrm{4}\left(\left(\mathrm{sin}\:{x}\:−\mathrm{sin}\:{t}\right)^{\mathrm{5}} \mathrm{sin}^{\mathrm{3}} \:{t}\right)^{\mathrm{1}/\mathrm{4}} }{\mathrm{sin}\:{x}\:\mathrm{cos}\:{t}}{du}\right] \\$$$$=\mathrm{4sin}\:{x}\:\int\frac{{u}^{\mathrm{6}} }{\left({u}^{\mathrm{4}} +\mathrm{1}\right)^{\mathrm{2}} }{du}= \\$$$$\:\:\:\:\:\left[\mathrm{Ostrogradski}\right] \\$$$$=\mathrm{sin}\:{x}\:\left(−\frac{{u}^{\mathrm{3}} }{{u}^{\mathrm{4}} +\mathrm{1}}+\mathrm{3}\int\frac{{u}}{{u}^{\mathrm{4}} +\mathrm{1}}{du}\right)= \\$$$$\:\:\:\:\:\left[\mathrm{3}\int\frac{{u}}{{u}^{\mathrm{4}} +\mathrm{1}}{du}=\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{4}}\int\left(\frac{{u}}{{u}^{\mathrm{2}} −\sqrt{\mathrm{2}}{u}+\mathrm{1}}−\frac{{u}}{{u}^{\mathrm{2}} +\sqrt{\mathrm{2}}{u}+\mathrm{1}}\right){du}\right] \\$$$$=\mathrm{sin}\:{x}\:\left(−\frac{{u}^{\mathrm{3}} }{{u}^{\mathrm{4}} +\mathrm{1}}+\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{8}}\left(\mathrm{ln}\:\frac{{u}^{\mathrm{2}} −\sqrt{\mathrm{2}}{u}+\mathrm{1}}{{u}^{\mathrm{2}} +\sqrt{\mathrm{2}}{u}+\mathrm{1}}\:+\mathrm{2}\left(\mathrm{arctan}\:\left(\sqrt{\mathrm{2}}{u}−\mathrm{1}\right)\:+\mathrm{arctan}\:\left(\sqrt{\mathrm{2}}{u}+\mathrm{1}\right)\right)\right)\right) \\$$$$... \\$$$${t}=\mathrm{0}\:\Rightarrow\:{u}=\mathrm{0} \\$$$${t}={x}\:\Rightarrow\:{u}=+\infty\:\left(?\right) \\$$$$\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{3}\pi\sqrt{\mathrm{2}}}{\mathrm{4}}\mathrm{sin}\:{x}\:\left(?\right) \\$$

Commented by benjo_mathlover last updated on 26/Jan/21

$${amazing} \\$$