SUMMARY
The Laplace transformation of cos²(t) can be computed using the integral formula \(\mathcal{L}(s) = \int_0^{\infty} e^{-st} \cos^2(t) \, dt\). This integral simplifies to \(\frac{1}{2} \int_0^{\infty} e^{-st} (\cos(2t) + 1) \, dt\), which can be further broken down into two separate integrals. The second integral, \(\int_0^{\infty} e^{-st} \, dt\), evaluates to \(-\frac{1}{s}\) when limits are applied from infinity to zero, providing a definitive solution for the Laplace transformation.
PREREQUISITES
- Understanding of Laplace transformations
- Familiarity with integral calculus
- Knowledge of trigonometric identities, specifically cos²(t)
- Basic proficiency in handling limits in calculus
NEXT STEPS
- Study the properties of Laplace transformations
- Learn about the use of trigonometric identities in integration
- Explore the application of Laplace transformations in solving differential equations
- Investigate advanced techniques for evaluating improper integrals
USEFUL FOR
Students studying engineering mathematics, mathematicians focusing on integral transforms, and anyone interested in applying Laplace transformations to solve differential equations.