### Video Transcript

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.