Pomoc za integral (zadatak)

[Matematicar]

Molim Vas da mi rjesite integral

integral x^2/(1+x^2 cosy) dx

Molim Vas da mi rjesite ovaj integral detaljno i sto prije....

Unaprijed hvalaa

[goranm]

Neka je a=cosy.

[dtex]\frac{x^2}{1+ax^2}=\frac{1}{a}\cdot\frac{x^2}{x^2+\frac{1}{a}}=\frac{1}{a}\cdot\frac{x^2+\frac{1}{a}-\frac{1}{a}}{x^2+\frac{1}{a}}=\frac{1}{a}\cdot\left(1-\frac{\frac{1}{a}}{x^2+\frac{1}{a}}\right)=\frac{1}{a}-\frac{1}{a^2}\cdot\frac{1}{x^2+\frac{1}{a}}[/dtex]

[dtex]\int\frac{x^2}{1+ax^2}dx=\frac{1}{a}x-\frac{1}{a^2}\int\frac{1}{x^2+\frac{1}{a}}dx=\frac{1}{a}x-\frac{\sqrt{a}}{a^2}\arctan(\sqrt{a}\cdot x)+C.[/dtex]

[dtex]\int\frac{x^2}{1+\cos{y}\cdot x^2}dx=\frac{x}{\cos{y}}-\frac{\arctan(\sqrt{\cos{y}}\cdot x)}{\cos^{\frac{3}{2}}y}+C.[/dtex]