# Question Video: Integrating Trigonometric Functions Involving Reciprocal Trigonometric Functions Mathematics

Determine ∫(9 tan 3𝑥 sec 3𝑥)d𝑥.

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### Video Transcript

Determine the indefinite integral of nine tan three 𝑥 sec three 𝑥 with respect to 𝑥.

The integrand here contains the product of a tangent and a secant, both of which have the same argument three 𝑥. We can recall the following standard integral. The indefinite integral of sec 𝑥 tan 𝑥 with respect to 𝑥 is equal to sec of 𝑥 plus 𝐶. To use this result in this example, we need to modify the argument three 𝑥 of the trigonometric functions. We can do this by making a substitution. We’ll let 𝑢 equal three 𝑥.

It follows then that d𝑢 by d𝑥 is equal to three. And whilst d𝑢 by d𝑥 is not a fraction, we can treat it a little like one. So equivalently d𝑢 is equal to three d𝑥, or one-third d𝑢 is equal to d𝑥. We can now perform this substitution replacing three 𝑥 with 𝑢 and d𝑥 with one-third d𝑢 to obtain the indefinite integral of nine tan 𝑢 sec 𝑢 one-third d𝑢. The constant in the integrand simplifies to three, and we can then bring this factor of three out the front to give three multiplied by the indefinite integral of tan 𝑢 sec 𝑢 with respect to 𝑢.

By the standard result, this integrates to three sec 𝑢 plus a constant of integration 𝐶. All that remains is to undo our substitution, so we need to replace 𝑢 with three 𝑥, which gives our final answer to the problem. We’ve found that the indefinite integral of nine tan three 𝑥 sec three 𝑥 with respect to 𝑥 is equal to three sec three 𝑥 plus a constant of integration 𝐶.