without-use-intergration-by-party-0-pi-4-e-cos-2-d-

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

w i t h o u t u s e i n t e r g r a t i o n

b y p a r t y

\int_{0}^{\frac{π}{4}} e^{θ} \cos 2 θ d θ

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

\int_{0}^{\frac{π}{4}} e^{θ} c o s (2 θ) d θ = R e (\int_{0}^{\frac{π}{4}} e^{θ + 2 i θ} d θ) a n d

\int_{0}^{\frac{π}{4}} e^{(1 + 2 i) θ} d θ = {[\frac{1}{1 + 2 i} e^{(1 + 2 i) θ}]}_{0}^{\frac{π}{4}} = \frac{1}{1 + 2 i} {e^{(1 + 2 i) \frac{π}{4}} - 1}

= \frac{(1 - 2 i)}{5} (e^{\frac{π}{4}} i - 1) = \frac{1}{5} (e^{\frac{π}{4}} i - 1 + 2 e^{\frac{π}{4}} + 2 i) \Rightarrow

\int_{0}^{\frac{π}{4}} e^{θ} c o s (2 θ) d θ = \frac{2 e^{\frac{π}{4}} - 1}{5}