Integration Questions

Question Number 145776 by Engr_Jidda last updated on 08/Jul/21

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{x}} {cosxdx} \\$$

Answered by ArielVyny last updated on 08/Jul/21

$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{x}} {cosxdx} \\$$$${du}={e}^{{x}} \rightarrow{u}={e}^{{x}} \\$$$${v}={cosx}\rightarrow{dv}=−{sinx} \\$$$${I}=\left[{e}^{{x}} {cosx}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} +{J}\:\:\left({J}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{x}} {sinxdx}\right) \\$$$${J}=\left[{e}^{{x}} {sinx}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} −{I} \\$$$$\begin{cases}{{I}−{J}=−\mathrm{1}}\\{{I}+{J}={e}^{\frac{\pi}{\mathrm{2}}} }\end{cases} \\$$$$\mathrm{2}{I}={e}^{\frac{\pi}{\mathrm{2}}} −\mathrm{1}\rightarrow{I}=\frac{{e}^{\frac{\pi}{\mathrm{2}}} −\mathrm{1}}{\mathrm{2}} \\$$$$\\$$

Answered by mathmax by abdo last updated on 08/Jul/21

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{e}^{\mathrm{x}} \:\mathrm{cosx}\:\mathrm{dx}\:=\mathrm{Re}\left(\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{e}^{\mathrm{x}+\mathrm{ix}} \mathrm{dx}\right)=\mathrm{Re}\left(\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{e}^{\left(\mathrm{1}+\mathrm{i}\right)\mathrm{x}} \mathrm{dx}\right)\:\mathrm{and} \\$$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{e}^{\left(\mathrm{1}+\mathrm{i}\right)\mathrm{x}} \:\mathrm{dx}\:=\left[\frac{\mathrm{1}}{\mathrm{1}+\mathrm{i}}\mathrm{e}^{\left(\mathrm{1}+\mathrm{i}\right)\mathrm{x}} \right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{i}}\left(\mathrm{e}^{\left(\mathrm{1}+\mathrm{i}\right)\frac{\pi}{\mathrm{2}}} −\mathrm{1}\right) \\$$$$=\frac{\mathrm{1}−\mathrm{i}}{\mathrm{2}}\left(\mathrm{i}\:\mathrm{e}^{\frac{\pi}{\mathrm{2}}} −\mathrm{1}\right)\:=\frac{\mathrm{ie}^{\frac{\pi}{\mathrm{2}}} −\mathrm{1}+\mathrm{e}^{\frac{\pi}{\mathrm{2}}} \:+\mathrm{i}}{\mathrm{2}}\:=\frac{\mathrm{e}^{\frac{\pi}{\mathrm{2}}} −\mathrm{1}}{\mathrm{2}}+\mathrm{i}\left(...\right)\:\Rightarrow \\$$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{e}^{\mathrm{x}} \mathrm{cosxdx}=\frac{\mathrm{e}^{\frac{\pi}{\mathrm{2}}} −\mathrm{1}}{\mathrm{2}} \\$$