- #1

- 126

- 0

**1. [itex]∫\frac{\sqrt{x^{2}-4}}{x} dx[/itex], [itex]x=2secθ[/itex], [itex]dx=2secθtanθ dθ[/itex]**

**2. [itex]\sqrt{x^{2}-a^{2}}[/itex],[itex]sec^{2}θ-1=tan^{2}θ[/itex]**

**3. [itex]\sqrt{x^{2}-4}=\sqrt{4sec^{2}θ-4}=\sqrt{4(1+tan^{2}θ)-4}=\sqrt{4tan^{2}θ}=2\left|tanθ\right|=2tanθ[/itex];[itex]∫\frac{\sqrt{x^{2}-4}}{x}dx=∫\frac{2tanθ}{2secθ}dθ=\frac{ln\left|secθ\right|}{ln\left|secθ+tanθ\right|}+C=ln\left|secθ\right|-ln\left|secθ+tanθ\right|+C[/itex];[itex]∫\frac{\sqrt{x^{2}-4}}{x}dx=ln\left|\frac{x}{2}\right|-ln\left|\frac{x}{2}+\frac{\sqrt{x^{2}-4}}{2}\right|+C=ln\left|x\right|-ln\left|2\right|-ln\left|x+\sqrt{x^{2}-4}\right|+ln\left|2\right|+C[/itex]. Please tell me what I did wrong.**