Задача 39460 Найти интеграл: ...

sat12

2019-09-14 21:55:53

Найти интеграл:

sova

2019-09-14 22:19:57

★

1.

[m]\int \frac{4\sqrt{1-x^2}+3x^2}{x^2-1}dx=-\int \frac{4\sqrt{1-x^2}}{1-x^2}dx+\int \frac{3x^2}{x^2-1}dx=[/m]

[m]=-4\int \frac{1}{\sqrt{1-x^2}}dx+3\int \frac{(x^2-1)+1}{x^2-1}dx=[/m]

[m]=-4\int \frac{1}{\sqrt{1-x^2}}dx+3\int \frac{(x^2-1)}{x^2-1}dx+3\int \frac{1}{x^2-1}dx=[/m]

[m]=-4arcsinx+3x+\frac{3}{2}ln|\frac{x-1}{x+1}|+C[/m]

2.


cos2x=cos^2x-sin^2x

[m]\int \frac{cos2x}{sin^2x\cdot cos^2x}dx=\int \frac{cos^2x-sin^2x}{sin^2x\cdot cos^2x}dx=[/m]

[m]=\int \frac{cos^2x}{sin^2x\cdot cos^2x}dx-\int \frac{sin^2x}{sin^2x\cdot cos^2x}dx=[/m]


[m]=\int \frac{1}{sin^2x}dx-\int \frac{1}{cos^2x}dx=-ctgx-tgx+C[/m]