SUMMARY
The integral simplification process for the expression $$\int \frac{x^2}{x^2 + 9} \,dx$$ involves rewriting the integrand as $$1 - \frac{9}{x^2 + 9}$$. This allows the integral to be separated into two simpler components: $$\int dx$$ and $$-9\int \frac{1}{x^2 + 9} \,dx$$. The first integral evaluates to $$x$$, while the second integral can be solved using the arctangent function, leading to a complete solution for the original integral.
PREREQUISITES
- Understanding of integral calculus
- Familiarity with algebraic manipulation of fractions
- Knowledge of basic integration techniques
- Experience with the arctangent function and its integral
NEXT STEPS
- Study the properties of definite and indefinite integrals
- Learn integration techniques involving substitution and partial fractions
- Explore the integral of the arctangent function
- Practice simplifying complex rational functions before integration
USEFUL FOR
Students and educators in calculus, mathematicians focusing on integral calculus, and anyone seeking to improve their skills in simplifying and solving integrals.