I'm reviewing my quizzes to study for midterm tomorrow, and I came across a problem where I'm supposed to integrate:
$$\int\frac{1}{x^2\sqrt{4-x^2}}dx$$
I used Mathematica to solve the problem and I'm sure it gave me the correct answer, which is: $$-\frac{\sqrt{4-x^2}}{4x}$$
I used $ x = 2\sin{\theta}$ and $dx = 2\cos{\theta}$ $d\theta$ to solve the problem, and I only got to $$\frac{1}{8}\int{\frac{1}{\sin^2{\theta}\cos{\theta}}}d\theta$$ Looking at step-by-step solution via WolframAlpha, they used $\theta = \arcsin{\frac{x}{2}}$ to solve the problem which I do not know how to. I don't think there is a need for $\theta = \arcsin{\frac{x}{2}}$ to solve the problem, and I'm wondering if anyone can show me how to solve this step by step without the use of $\theta = \arcsin{\frac{x}{2}}$? Maybe help me understand how to?
Trigonometric substitution is the only method that I'm struggling with, and any tips on improving trig sub skill would be appreciated too.
Thanks.