Question Number 29105 by tawa tawa last updated on 04/Feb/18
$$\mathrm{Show}\:\mathrm{that}:\:\:\:\int_{−\mathrm{1}} ^{\:\:\:\mathrm{1}} \:\:\:\:\:\:\:\frac{\mathrm{dx}}{\mathrm{5}\:\mathrm{cosh}\left(\mathrm{x}\right)\:+\:\mathrm{13}\:\mathrm{sinh}\left(\mathrm{x}\right)}\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{log}_{\mathrm{e}} \left(\frac{\mathrm{15e}\:−\:\mathrm{10}}{\mathrm{3e}\:+\:\mathrm{2}}\right)\: \\ $$
Commented by abdo imad last updated on 04/Feb/18
$${I}=\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\:\:\:\:\:\:\:\:\:\frac{{dx}}{\mathrm{5}\frac{{e}^{{x}} \:+{e}^{−{x}} }{\mathrm{2}}\:+\mathrm{13}\:\frac{{e}^{{x}} \:−{e}^{−{x}} }{\mathrm{2}}} \\ $$$$=\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\:\:\:\:\:\:\:\:\frac{\mathrm{2}{dx}}{\mathrm{18}\:{e}^{{x}} \:−\mathrm{8}\:{e}^{−{x}} }\:{thech}.\:{e}^{{x}} ={t}\:{give} \\ $$$${I}=\:\int_{{e}^{−\mathrm{1}} } ^{{e}} \:\:\:\:\:\:\:\:\frac{\mathrm{2}}{\mathrm{18}{t}\:−\frac{\mathrm{8}}{{t}}}\:\frac{{dt}}{{t}}=\:\int_{{e}^{−\mathrm{1}} } ^{{e}} \:\:\:\:\:\frac{\mathrm{2}}{\mathrm{18}{t}^{\mathrm{2}} \:−\mathrm{8}}{dt} \\ $$$${I}=\:\:\int_{{e}^{−\mathrm{1}} } ^{{e}} \:\:\:\:\:\frac{{dt}}{\mathrm{9}{t}^{\mathrm{2}} \:−\mathrm{4}}\:\:\:=>\:\:\mathrm{9}{I}=\frac{\mathrm{3}}{\mathrm{4}}\:\int_{{e}^{−\mathrm{1}} } ^{{e}} \:\left(\:\frac{\mathrm{1}}{{t}−\frac{\mathrm{2}}{\mathrm{3}}}\:−\:\frac{\mathrm{1}}{{t}+\frac{\mathrm{2}}{\mathrm{3}}}\right){dt} \\ $$$${I}=\:\frac{\mathrm{1}}{\mathrm{12}}\:{ln}\mid\:\frac{{t}−\frac{\mathrm{2}}{\mathrm{3}}}{{t}+\frac{\mathrm{2}}{\mathrm{3}}}\mid_{{e}^{−\mathrm{1}} } ^{{e}} =\:\frac{\mathrm{1}}{\mathrm{12}}\left({ln}\mid\:\frac{\mathrm{3}{e}−\mathrm{2}}{\mathrm{3}{e}\:+\mathrm{2}}\mid\:−{ln}\mid\frac{\mathrm{3}{e}^{−\mathrm{1}} −\mathrm{2}}{\mathrm{3}{e}^{−\mathrm{1}} \:+\mathrm{2}}\mid….\right. \\ $$$$ \\ $$
Commented by tawa tawa last updated on 04/Feb/18
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$