 Let's continue with our examples and look at some anti-derivatives that are going to involve trig functions. Here we have the anti-derivative of 3 secant x tangent x minus 5 cosecant squared x dx. There is no need to rewrite this integrand because we do have anti-derivative rules both for secant x tangent x and for cosecant squared of x. The three will remain as our coefficient. Think back to the rules we just talked about. Anti-derivative of secant of x tangent of x is secant of x minus, now we keep the 5 as our coefficient. Anti-derivative of cosecant squared of x is negative cotangent x and don't forget our constant of integration. We can then rewrite this very easily. As we've seen before, please do take a moment to take the derivative of your answer and you should notice you get the integrand back again. Derivative of secant is secant tangent. Derivative of cotangent is negative cosecant squared, so you can be assured that you're getting these correct. This one will require us to rewrite it and sometimes you have to be a little creative because you only have so many rules that you know, therefore you need to take what you're given and turn it into something that matches up with those rules. So in this one, we have two cotangent of x minus three sine squared of x all over sine of x dx. If we were to bring that sine of x up to the numerator to join the cotangent of x, it would turn into anti-derivative of two cotangent of x. When we bring the sine up, remember that becomes cosecant of x and we do have a rule for cotangent cosecant. Then we'd have minus three sine squared of x over sine of x. We can cancel one of the sines and of course we do have an anti-derivative rule for sine of x. The tube remains. Anti-derivative of cotangent of x, cosecant of x is negative cosecant x. I'm going to put the negative in front of the two. Anti-derivative of sine is negative cosine. Don't forget your constant of integration. Our final answer then is negative two cosecant of x plus three cosine of x plus C. Once again, you can check your answer by taking the derivative. You should get this into grand back again now. Of course, this also presupposes that you did that rewriting from the beginning correctly. Here we have anti-derivative of tangent squared of x plus cotangent squared of x plus four. Well, we do not have any rules for the anti-derivative of tangent squared or cotangent squared. So this is a case in which we're going to have to use our Pythagorean identities in order to write it first. Tangent squared is equal to secant squared of x minus one. Cotangent squared is equal to cosecant squared of x minus one and then we have the plus four. Let's go ahead and simplify that a little bit before we actually do our anti-derivative. Negative one minus one plus four of course is plus two. It saves us a little bit of work. Notice how we're keeping the differential there as we rewrite this. Now we have rules for these. Anti-derivative of secant squared of x is tangent of x. Anti-derivative of cosecant squared of x is negative cotangent x. Anti-derivative of two would be 2x. Once again, take the derivative of that. You'll notice you get your integrand back again. And we have one last example. Anti-derivative of four minus three over one plus x squared dx. Going term by term, anti-derivative of four is simply 4x. For the second part, this is where you need to recognize that integrand pattern. That is the pattern for an inverse tangent. So we'd have minus three, the three remains, as our coefficient. You can either write it as arc tangent or tangent to the negative one of x. Doesn't matter. Plus C. Again, take your derivative. You'll notice you get that integrand back again.