The integral of cot^6(x) can be computed without using the reduction formula by applying trigonometric identities. Specifically, the identity cot^2(x) = csc^2(x) - 1 allows for the expression of cot^6(x) as cot^4(x)(csc^2(x) - 1). This leads to the expansion cot^4(x) * csc^2(x) - cot^4(x), which can be integrated using substitution. By letting u = cot(x) and du = -csc^2(x), both terms can be integrated effectively.
PREREQUISITESStudents studying calculus, particularly those focusing on integration techniques, as well as educators looking for effective methods to teach trigonometric integrals.
Use the identity cot^2(x) = csc^2(x) - 1 to write cot^6(x) as cot^4(x)(csc^2(x) - 1). Expanding this gives you cot^4(x) * csc^2(x) - cot^4(x).rdioface said:Homework Statement
Find the integral of cot^6(x) without using the reduction formula.
Homework Equations
Potentially any trig identities involving the cotangent
The Attempt at a Solution
I tried splitting up the cot^6 in various ways such as cot^4*cot^2 and cot^2*cot^2*cot*2 but nothing has produced a solution, nor has doing a similar splitting and integrating by parts.