SUMMARY
The integral $\int {\frac{sin^{2}x}{1+sin^{2}x}dx}$ can be solved using the substitution \( t = \tan \frac{x}{2} \), leading to the transformed integral \( \int {\frac{8t^{2}}{(1+6t^{2}+t^{4})(1+t^{2})} dt} \). After applying partial fractions, the integral separates into two parts: \( \int {\frac{2}{1+t^{2}} dt} \) and \( \int {\frac{-2t^{2}-2}{(1+6t^{2}+t^{4})} dt} \). The first part simplifies to \( 2 \tan^{-1}(t) \), while the second requires further analysis to identify its integral. The discussion confirms that partial fractions are essential for progressing through the solution.
PREREQUISITES
- Understanding of trigonometric identities and integrals
- Familiarity with the substitution method in calculus
- Knowledge of partial fraction decomposition
- Ability to integrate rational functions
NEXT STEPS
- Study the method of integration by substitution in detail
- Learn about partial fraction decomposition techniques
- Explore the integration of rational functions involving polynomials
- Investigate the properties and applications of the arctangent function
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, as well as educators seeking to enhance their understanding of trigonometric integrals and substitution methods.
