# Video: Integrating Trigonometric Functions Involving Reciprocal Trigonometric Functions

Determine ∫ (−3 tan² 8𝑥 csc² 8𝑥) d𝑥.

01:29

### Video Transcript

Determine the indefinite integral of negative three tan squared eight 𝑥 times csc squared eight 𝑥 evaluated with respect to 𝑥. These does at first look quite tricky. However, If we recall some of our trigonometry identities, it does get a little nicer. We know that tan 𝑥 is equal to sin 𝑥 over cos of 𝑥. And we also know that csc 𝑥 is equal to one over sin 𝑥. We can, therefore, rewrite our entire integrant as negative three times sin squared eight 𝑥 over cos squared eight 𝑥 times one over sin squared eight 𝑥. And then we noticed that the sin squared eight 𝑥 cancels. We can take the factor of negative three outside of the integral sin to make the next step easier. And we have a negative three times the integral of one over cos squared eight 𝑥 d𝑥.

But we know that one over cos of 𝑥 is equal to sec of 𝑥. So our integral becomes negative three times the integral of sec squared eight 𝑥 evaluated with respect to 𝑥. But of course, the integral of sec squared 𝑎𝑥, evaluated with respect to 𝑥, is one over 𝑎 tan of 𝑎𝑥 plus some constant of integration 𝑐. And so we see that the integral of sec squared eight 𝑥 is an eighth tan of eight 𝑥 plus 𝑐. We distribute our parentheses. And we see that we’re left with negative three-eights of tan of eight 𝑥 plus a new constant since we multiplied our original one by negative three. Let’s call that capital 𝐶.