Помогите найти интеграл[tex] \frac{ \sqrt{1 + \tan(x) } }{ { \cos(x) }^{2} } dx[/tex]. Created by shonikster. matematika-ru

Ответ: Пошаговое объяснение:

tex]...

shonikster @shonikster

July 2022 1 4 Report

Помогите найти интеграл

[tex] \frac{ \sqrt{1 + \tan(x) } }{ { \cos(x) }^{2} } dx[/tex]

Answers & Comments

Helper211

Ответ: $\frac{2(1+tgx)^{\frac{3}{2} }}{3} +C$

Пошаговое объяснение:

$\int {\frac{\sqrt{1+tg(x)} }{cos^2(x)} dx}=\int {\sqrt{1+tgx}} \, d(tgx)$

$t=tgx\\\\dt=d(tgx)$

$\int {\sqrt{1+tgx}} \, d(tgx)=\int {\sqrt{1+t}} \, dt\\\\u=1+t\\\\du=dt\\\\\int {\sqrt{1+t}} \, dt=\int {\sqrt{u}} \, du=\frac{2u^{\frac{3}{2} }}{3} +C=\frac{2(1+tgx)^{\frac{3}{2} }}{3} +C$

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