問題: I = ∫_{0}^{π/2} e^x (1 + sinx)/(1 + cosx) dx を求めよ

問題: I = ∫_{0}^{π/2} e^x (1 + sinx)/(1 + cosx) dx を求めよat MATH

問題: I = ∫_{0}^{π/2} e^x (1 + sinx)/(1 + cosx) dx を求めよ - 暇つぶし2ch4:１３２人目の素数さん
25/08/05 08:33:39.21 GPzqfPLK.net
f(x) = e^t (1 + sinx)/(1 + cosx)

x = 2tとおくと
dx = 2dt
積分区間は0≤t≤π/4

f(x)dx
= e^2t (1 + 2cost sint)/2(cost)^2 2dt
= e^2t 1/(cost)^2 + 2e^2t tant
= e^2t (tant)' + (e^2t)' tant
= (e^2t tant)'

∴ I = e^(π/2)

次ページ