Question Number 213948 by issac last updated on 22/Nov/24
$$\mathrm{evaluate}. \\ $$$$\mathrm{1}.\:\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:\:{e}^{−\boldsymbol{{i}}\left({t}−\mathrm{sin}\left({t}\right)\right)} \mathrm{d}{t} \\ $$$$\mathrm{2}.\:\int_{\mathrm{0}} ^{\:\mathrm{a}} \int_{\mathrm{0}} ^{\:\mathrm{a}} \:\:\sqrt{{u}^{\mathrm{2}} +{v}^{\mathrm{2}} −\mathrm{6}{u}+\mathrm{9}}\:\mathrm{d}{u}\mathrm{d}{v} \\ $$$$\mathrm{3}.\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \:\:{e}^{\mathrm{cos}\left({t}\right)} \mathrm{cos}\left(\mathrm{2}{t}+\mathrm{sin}\left({t}\right)\right)\mathrm{d}{t} \\ $$$$\mathrm{4}.\:\int_{−\infty} ^{\:\infty} \:\frac{\mathrm{sin}\left(\mathrm{3}{z}\right)}{{z}^{\mathrm{2}} +\mathrm{2}{z}+\mathrm{5}}\:\mathrm{d}{z} \\ $$$$\mathrm{5}.\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:\:\frac{\mathrm{1}}{\mathrm{2}+\mathrm{cos}\left(\theta\right)}\:\mathrm{d}\theta \\ $$