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Question Number 89560 by peter frank last updated on 18/Apr/20

$$withoutuseintergration$$$$byparty$$$${\int }_0^{\frac{\pi }{4}}{e}^{\theta }\cos 2\theta d\theta$$

Commented by mathmax by abdo last updated on 18/Apr/20

$${\int }_0^{\frac{\pi }{4}}{e}^{\theta }cos\left(2\theta \right)d\theta =Re\left({\int }_0^{\frac{\pi }{4}}{e}^{\theta +2i\theta }d\theta \right)and$$$${\int }_0^{\frac{\pi }{4}}{e}^{(1+2i)\theta }d\theta ={\left[\frac{1}{1+2i}{e}^{(1+2i)\theta }\right]}_0^{\frac{\pi }{4}}=\frac{1}{1+2i}\{e^{(1+2i)\frac{\pi }{4}}-1\}$$$$=\frac{(1-2i)}{5}(e^{\frac{\pi }{4}}i-1)=\frac{1}{5}(e^{\frac{\pi }{4}}i-1+2e^{\frac{\pi }{4}}+2i)\Rightarrow$$$${\int }_0^{\frac{\pi }{4}}{e}^{\theta }cos\left(2\theta \right)d\theta =\frac{2e^{\frac{\pi }{4}}-1}{5}$$

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