Determine the indefinite integral
of two csc three 𝑥 cot three 𝑥 with respect to 𝑥.
The given integrand is the product
of a cosecant and cotangent function, both with the argument of three 𝑥. We recall the standard integral of
the product of the cosecant and cotangent functions. The indefinite integral of csc of
𝑥 multiplied by cot of 𝑥 with respect to 𝑥 is equal to the negative csc of 𝑥
plus 𝐶. In the given integrand, the
argument for both functions is three 𝑥 rather than 𝑥. So in order to apply the standard
integral, we need to use a substitution.
We let 𝑢 equal three 𝑥, which in
turn implies that d𝑢 by d𝑥 is equal to three. And so one-third d𝑢 is equivalent
to d𝑥. Making this change of variable in
the integral, we get the indefinite integral of two csc 𝑢 cot 𝑢 one-third d𝑢. Taking the constant factor of
two-thirds out the front of the integral and then applying our standard result gives
negative two-thirds csc 𝑢 plus a constant of integration 𝐶. All that remains is to reverse the
substitution by replacing 𝑢 with three 𝑥.
And so we obtain our final
answer. The indefinite interval of two csc
three 𝑥 cot three 𝑥 with respect to 𝑥 is equal to negative two-thirds csc of
three 𝑥 plus 𝐶.