Integration Questions

Question Number 101272 by bemath last updated on 01/Jul/20

$$\int\sqrt{\mathrm{sec}\:{x}}\:{dx}\: \\$$

Commented by john santu last updated on 01/Jul/20

$$\mathrm{I}=\:\int\frac{{dx}}{\sqrt{\mathrm{cos}\:{x}}}\:=\:\int\:\frac{\mathrm{1}+\mathrm{sin}\:{x}−\mathrm{sin}\:{x}}{\sqrt{\mathrm{cos}\:{x}}}\:{dx} \\$$$$\mathrm{I}_{\mathrm{1}} \:=\:\int\:\frac{\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}−\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)^{\mathrm{2}} \:{dx}}{\sqrt{\mathrm{cos}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{sin}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}} \\$$$$\mathrm{I}_{\mathrm{1}} \:=\:−\mathrm{2}\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}+\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \\$$$$\mathrm{I}_{\mathrm{2}} \:=\:−\int\:\frac{\mathrm{sin}\:{x}}{\sqrt{\mathrm{cos}\:{x}}}\:{dx}\:=\:\int\:\frac{{d}\left(\mathrm{cos}\:{x}\right)}{\sqrt{\mathrm{cos}\:{x}}} \\$$$$=\:\mathrm{2}\sqrt{\mathrm{cos}\:{x}} \\$$$$\mathrm{I}\:=\:\mathrm{I}_{\mathrm{1}} \:+\:\mathrm{I}_{\mathrm{2}} \:=\:\mathrm{2}\sqrt{\mathrm{cos}\:{x}}−\mathrm{2}\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}+\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} +\:{c}\: \\$$

Commented by MJS last updated on 01/Jul/20

$$\mathrm{sorry}\:\mathrm{but}\:\mathrm{I}_{\mathrm{1}} \neq−\mathrm{2}\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\:+\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)^{\mathrm{3}/\mathrm{2}} \\$$$$\mathrm{because}\:\frac{\left({c}−{s}\right)^{\mathrm{2}} }{\sqrt{{c}^{\mathrm{2}} −{s}^{\mathrm{2}} }}=\frac{\left({c}−{s}\right)^{\mathrm{3}/\mathrm{2}} }{\sqrt{{c}+{s}}}\:\mathrm{and}\:\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{see} \\$$$$\mathrm{how}\:\mathrm{to}\:\mathrm{get}\:\mathrm{your}\:\mathrm{solution} \\$$$$\\$$

Answered by mathmax by abdo last updated on 01/Jul/20

$$\mathrm{this}\:\mathrm{integral}\:\mathrm{is}\:\mathrm{not}\:\mathrm{solvable}\:\left(\mathrm{elliptic}\right) \\$$

Commented by MJS last updated on 01/Jul/20

$$\mathrm{yes}\:\mathrm{you}\:\mathrm{are}\:\mathrm{right} \\$$