SUMMARY
The integral of the function \(\frac{x}{x^2 + 1}\) from 1 to infinity can be solved using the substitution \(u = x^2 + 1\), leading to \(du = 2x \, dx\) and \(\frac{du}{2} = x \, dx\). This substitution simplifies the integral to \(\frac{1}{2} \int \frac{1}{u} \, du\), which evaluates to \(\frac{1}{2} \ln|u|\). Applying the limit as \(x\) approaches infinity results in \(\frac{1}{2} \ln(\infty)\), indicating that the integral diverges to infinity.
PREREQUISITES
- Understanding of integral calculus
- Familiarity with substitution methods in integration
- Knowledge of limits in calculus
- Basic logarithmic functions and their properties
NEXT STEPS
- Study advanced techniques in integral calculus
- Learn about improper integrals and their convergence
- Explore logarithmic properties and their applications in calculus
- Investigate the concept of limits and their significance in calculus
USEFUL FOR
Students of calculus, mathematics educators, and anyone interested in advanced integration techniques and limit applications.