SUMMARY
The integral $$\int_{}^{} \frac {x^2}{{(4 - x^2)}^{3/2}}\,dx$$ can be solved using the trigonometric substitution $x = 2\sin\theta$. This substitution leads to $dx = 2\cos\theta d\theta$ and simplifies the expression for $\sqrt{4 - x^2}$ to $2\cos\theta$. The identity $\sin^2\theta + \cos^2\theta = 1$ is crucial for deriving this relationship, confirming that $\sqrt{4 - x^2} = 2\cos\theta$ is valid.
PREREQUISITES
- Understanding of trigonometric identities, specifically $\sin^2\theta + \cos^2\theta = 1$
- Familiarity with integral calculus and substitution methods
- Knowledge of trigonometric functions and their properties
- Basic skills in manipulating algebraic expressions
NEXT STEPS
- Study trigonometric substitution techniques in integral calculus
- Learn how to apply the identity $\sin^2\theta + \cos^2\theta = 1$ in various contexts
- Explore more complex integrals involving trigonometric functions
- Practice solving integrals using different substitution methods
USEFUL FOR
Students and professionals in mathematics, particularly those studying calculus, as well as educators teaching integral calculus and trigonometric substitutions.