Integration Questions

Question Number 161537 by cortano last updated on 19/Dec/21

$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{1}+\mathrm{tan}\:^{\mathrm{4}} \left({x}\right)}{\mathrm{cot}\:^{\mathrm{2}} \left({x}\right)}\:{dx}\:=? \\$$

Answered by Ar Brandon last updated on 19/Dec/21

$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left(\mathrm{tan}^{\mathrm{2}} {x}+\mathrm{tan}^{\mathrm{6}} {x}\right){dx} \\$$ $$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left(\mathrm{sec}^{\mathrm{2}} {x}−\mathrm{1}\right){dx}+\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left(\mathrm{tan}^{\mathrm{4}} {x}\right)\left(\mathrm{sec}^{\mathrm{2}} {x}−\mathrm{1}\right){dx} \\$$ $$=\left[\mathrm{tan}{x}−{x}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} +\left[\frac{\mathrm{tan}^{\mathrm{5}} {x}}{\mathrm{5}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} −\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left(\mathrm{tan}^{\mathrm{2}} {x}\right)\left(\mathrm{sec}^{\mathrm{2}} {x}−\mathrm{1}\right){dx} \\$$ $$=\mathrm{1}−\frac{\pi}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}}−\left[\frac{\mathrm{tan}^{\mathrm{3}} {x}}{\mathrm{3}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} +\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left(\mathrm{sec}^{\mathrm{2}} {x}−\mathrm{1}\right){dx} \\$$ $$=\frac{\mathrm{6}}{\mathrm{5}}−\frac{\pi}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{3}}+\left[\mathrm{tan}{x}−{x}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} =\frac{\mathrm{13}}{\mathrm{15}}−\frac{\pi}{\mathrm{4}}+\left(\mathrm{1}−\frac{\pi}{\mathrm{4}}\right) \\$$ $$=\frac{\mathrm{28}}{\mathrm{15}}−\frac{\pi}{\mathrm{2}} \\$$

Commented byAr Brandon last updated on 19/Dec/21

$$\mathrm{1}+\mathrm{tan}^{\mathrm{2}} {x}=\mathrm{sec}^{\mathrm{2}} {x} \\$$ $$\frac{{d}\left(\mathrm{tan}{x}\right)}{{dx}}=\mathrm{sec}^{\mathrm{2}} {x} \\$$

Commented bypeter frank last updated on 20/Dec/21

$$\mathrm{good} \\$$

Answered by cortano last updated on 19/Dec/21

Commented bysaboorhalimi last updated on 19/Dec/21

$${sir}\:{which}\:{software}\:{did}\:{you}\:{use}\: \\$$ $${for}\:{writing}\:{this}\:{solution}? \\$$

Commented bycortano last updated on 20/Dec/21

$${math}\:{editor}\:{for}\:{pc} \\$$