SUMMARY
The discussion focuses on evaluating the definite integral of the function \(\int \sin(3t) dt\) with boundaries from \(0\) to \(\frac{\pi}{3}\). The correct approach involves recognizing that the integral of \(\sin(nx)\) is given by the formula \(-\frac{\cos(nx)}{n} + C\). A substitution \(x = 3t\) leads to \(dt = \frac{1}{3}dx\), clarifying the integration process. The initial confusion stemmed from incorrectly stating that \(dt = 3\), which is not valid in differential calculus.
PREREQUISITES
- Understanding of definite integrals and their evaluation
- Familiarity with trigonometric functions and their integrals
- Knowledge of substitution methods in integration
- Basic concepts of differential calculus
NEXT STEPS
- Study the integration of trigonometric functions, specifically \(\int \sin(nx) dx\)
- Learn about substitution techniques in integral calculus
- Practice evaluating definite integrals with various functions
- Explore common mistakes in differential calculus and how to avoid them
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, as well as educators seeking to clarify common misconceptions in integral calculus.