Задача 67854 ...

628a022a6653d96aeebf61ed

23.12.2022 11:25:16

Вычислить интеграл ∫ sqrt(1+ sin^2x/cos^2x) от 0 до Pi/4

5f3ea7e3faf909182968ddd9

23.12.2022 11:33:26

★

[m] ∫_{0}^{\frac{π}{4}}\sqrt{ 1+\frac{sin^2x}{cos^2x}}dx=∫_{0}^{\frac{π}{4}}\sqrt{ \frac{cos^x+sin^2x}{cos^2x}}dx=∫_{0}^{\frac{π}{4}}\sqrt{ \frac{1}{cos^2x}}dx=∫_{0}^{\frac{π}{4}}\frac{1}{cosx}dx=[/m]

табличный

[m]=(ln|tg(\frac{x}{2}+\frac{π}{4})|)|_{0}^{\frac{π}{4}}=ln tg\frac{3π}{8}-lntg\frac{π}{4}=ln tg\frac{3π}{8}-ln1=ln tg\frac{3π}{8}[/m]