SUMMARY
The integral of sin^3(x)cos^6(x)dx is solved using the substitution method, where u = cos(x) and du = -sin(x)dx. The integral simplifies to -∫(u^63 - u^65)du, resulting in the final expression of (-cos^64(x)/64) - (cos^66(x)/66) + c. Verification through differentiation confirms the correctness of the solution, ensuring that the original integrand is recovered.
PREREQUISITES
- Understanding of integral calculus and substitution methods
- Familiarity with trigonometric identities and their derivatives
- Knowledge of integration techniques, particularly polynomial integration
- Ability to differentiate functions to verify integration results
NEXT STEPS
- Study advanced integration techniques, including integration by parts
- Explore trigonometric integrals and their simplifications
- Learn about the use of substitution in definite integrals
- Practice verifying integrals through differentiation with various functions
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of trigonometric integrals and substitution methods.