Question Number 57417 by Abdo msup. last updated on 03/Apr/19 | ||
$${let}\:{F}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$ | ||
Commented by Abdo msup. last updated on 05/Apr/19 | ||
$$\left.\mathrm{1}\right){changement}\:{tan}\left(\frac{{t}}{\mathrm{2}}\right)={u}\:{give} \\ $$$${F}\left({x}\right)=\int_{\mathrm{0}} ^{{tan}\left(\frac{{x}}{\mathrm{2}}\right)} \:\:\:\frac{\mathrm{1}+\frac{\mathrm{2}{u}}{\mathrm{1}+{u}^{\mathrm{2}} }}{\mathrm{2}+\frac{\mathrm{1}−{u}^{\mathrm{2}} }{\mathrm{1}+{u}^{\mathrm{2}} }}\:\frac{\mathrm{2}{du}}{\mathrm{1}+{u}^{\mathrm{2}} } \\ $$$$=\int_{\mathrm{0}} ^{{tan}\left(\frac{{x}}{\mathrm{2}}\right)} \:\:\:\:\:\frac{\mathrm{1}+{u}^{\mathrm{2}} \:+\mathrm{2}{u}}{\mathrm{2}+\mathrm{2}{u}^{\mathrm{2}} \:+\mathrm{1}−{u}^{\mathrm{2}} }\:\frac{\mathrm{2}{du}}{\mathrm{1}+{u}^{\mathrm{2}} } \\ $$$$=\mathrm{2}\:\int_{\mathrm{0}} ^{{tan}\left(\frac{{x}}{\mathrm{2}}\right)} \:\:\:\frac{{u}^{\mathrm{2}} \:+\mathrm{2}{u}\:+\mathrm{1}}{\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\left({u}^{\mathrm{2}} \:+\mathrm{3}\right)}\:{du}\:\:{let}\:{decompose} \\ $$$${F}\left({u}\right)\:=\:\frac{{u}^{\mathrm{2}} \:+\mathrm{2}{u}\:+\mathrm{1}}{\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\left({u}^{\mathrm{2}} \:+\mathrm{3}\right)}\:\Rightarrow{F}\left({u}\right)\:=\frac{{au}\:+{b}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{{cu}\:+{d}}{{u}^{\mathrm{2}} \:+\mathrm{3}} \\ $$$$\Rightarrow\left({u}^{\mathrm{2}} \:+\mathrm{3}\right)\left({au}+{b}\right)\:+\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\left({cu}\:+{d}\right)={u}^{\mathrm{2}} \:+\mathrm{2}{u}\:+\mathrm{1}\:\Rightarrow \\ $$$${au}^{\mathrm{3}} \:+{bu}^{\mathrm{2}} \:+\mathrm{3}{au}\:+\mathrm{3}{b}\:+{cu}^{\mathrm{3}} \:+{du}^{\mathrm{2}} \:+{cu}\:+{d}\:={u}^{\mathrm{2}} \:+\mathrm{2}{u}\:+\mathrm{1}\:\Rightarrow \\ $$$$\left({a}+{c}\right){u}^{\mathrm{3}} \:+\left({b}+{d}\right){u}^{\mathrm{2}} \:+\left(\mathrm{3}{a}+{c}\right){u}\:+\mathrm{3}{b}\:+{d}\:={u}^{\mathrm{2}} \:+\mathrm{2}{u}\:+\mathrm{1}\Rightarrow \\ $$$${a}+{c}=\mathrm{0},{b}+{d}\:=\mathrm{1},\mathrm{3}{a}+{c}\:=\mathrm{2}\:,\mathrm{3}{b}\:+{d}\:=\mathrm{1}\:\Rightarrow \\ $$$${c}=−{a}\:\Rightarrow\mathrm{3}{a}−{a}\:=\mathrm{2}\:\Rightarrow{a}\:=\mathrm{1}\:\:{wehave}\:{d}=\mathrm{1}−{b}\:\Rightarrow \\ $$$$\mathrm{3}{b}\:+\mathrm{1}−{b}\:=\mathrm{1}\:\Rightarrow{b}=\mathrm{0}\:\Rightarrow{d}=\mathrm{1}\:\Rightarrow \\ $$$${F}\left({u}\right)=\frac{{u}}{{u}^{\mathrm{2}} \:+\mathrm{1}}\:+\frac{−{u}\:+\mathrm{1}}{{u}^{\mathrm{2}} \:+\mathrm{3}}\:\Rightarrow\int\:{F}\left({u}\right){du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\:−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({u}^{\mathrm{2}} \:+\mathrm{3}\right)\:+{arctan}\left({u}\right)\:+{c}\:\Rightarrow \\ $$$${F}\left({x}\right)=\mathrm{2}\:\left[\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({u}^{\mathrm{2}} \:+\mathrm{3}\right)\:+{arctan}\left({u}\right)\right]_{\mathrm{0}} ^{{tan}\left(\frac{{x}}{\mathrm{2}}\right)} \\ $$$$=\mathrm{2}\:\left\{\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{tan}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)\right)−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{3}+{tan}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)\right)+\frac{{x}}{\mathrm{2}}\right\} \\ $$$${F}\left({x}\right)={x}\:+{ln}\left(\mathrm{1}+{tan}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)\right)−{ln}\left(\mathrm{3}+{tan}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)\right) \\ $$ | ||
Commented by Abdo msup. last updated on 05/Apr/19 | ||
$$\left.\mathrm{2}\right){we}\:{have}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt}\:=_{{tan}\left(\frac{{x}}{\mathrm{2}}\right)} \:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+\frac{\mathrm{2}{u}}{\mathrm{1}+{u}^{\mathrm{2}} }}{\mathrm{2}+\frac{\mathrm{1}−{u}^{\mathrm{2}} }{\mathrm{1}+{u}^{\mathrm{2}} }}\:\frac{\mathrm{2}{du}}{\mathrm{1}+{u}^{\mathrm{2}} } \\ $$$$=\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{u}^{\mathrm{2}} \:+\mathrm{2}{u}\:+\mathrm{1}}{\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)\left({u}^{\mathrm{2}} \:+\mathrm{3}\right)}\:{du} \\ $$$$=\left[{ln}\left({u}^{\mathrm{2}} \:+\mathrm{1}\right)−{ln}\left({u}^{\mathrm{2}} \:+\mathrm{3}\right)\:+\mathrm{2}\:{arctan}\left({u}\right)\right]_{\mathrm{0}} ^{+\infty} \\ $$$$=\left[{ln}\left(\frac{{u}^{\mathrm{2}} \:+\mathrm{1}}{{u}^{\mathrm{2}} \:+\mathrm{3}}\right)\:+\mathrm{2}\:{arctan}\left({u}\right)\right]_{\mathrm{0}} ^{+\infty} =\pi−{ln}\left(\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$=\pi\:+{ln}\left(\mathrm{3}\right)\:. \\ $$ | ||