SUMMARY
The integral of \( \frac{x^4}{(1-x^2)^{3/2}} \, dx \) can be solved using the substitution \( x = \sin(t) \), which leads to \( dx = \cos(t) \, dt \). This transforms the integral into \( \int \frac{\sin^4(t)}{\cos^2(t)} \, dt \). Further simplification involves rewriting \( \sin^4(t) \) as \( (1 - \cos(2t))^2 \) and expanding it for easier integration.
PREREQUISITES
- Understanding of integral calculus
- Familiarity with trigonometric identities
- Knowledge of substitution methods in integration
- Ability to manipulate and expand polynomial expressions
NEXT STEPS
- Study advanced techniques in integral calculus
- Learn about trigonometric substitution in integrals
- Explore polynomial expansions and their applications in integration
- Investigate the use of right triangles in trigonometric integrals
USEFUL FOR
Students and professionals in mathematics, particularly those focusing on calculus and integral techniques, as well as educators looking for methods to teach integration strategies.