SUMMARY
The integral of the function \(\int \sec^{2}(3x)e^{\tan(3x)}dx\) can be solved using the substitution method. By letting \(u = \tan(3x)\), the integral simplifies to \(\int e^{u} \cdot \sec^{2}(3x)dx\). The differential \(dx\) is expressed as \(dx = \frac{du}{\sec^{2}(3x)} \cdot \frac{1}{3}\), leading to the final solution of \(\frac{1}{3}e^{\tan(3x)} + C\), confirming the correctness of the approach.
PREREQUISITES
- Understanding of integration techniques, specifically integration by substitution.
- Familiarity with trigonometric identities, particularly \(\sec^{2}(x) = 1 + \tan^{2}(x)\).
- Knowledge of exponential functions and their derivatives.
- Ability to manipulate differentials in calculus.
NEXT STEPS
- Practice more problems involving integration by substitution with trigonometric functions.
- Explore the relationship between derivatives and integrals in the context of exponential functions.
- Study the application of trigonometric identities in calculus.
- Learn about advanced integration techniques such as integration by parts and partial fractions.
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of substitution in integrals.