 Here we're going to talk about anti-derivatives involving our trigonometric functions. These essentially become more rules you need to memorize. But if you know your derivative rules really well, these are not so bad. We're going to use the fact that we know derivatives and anti-derivatives are inverses of each other. So if we want the anti-derivative of sine of x dx, think to yourself, what can I take the derivative of and get sine as my answer? Hopefully you've thought of the fact, if you did the derivative of negative cosine of x plus c, you get sine of x back. Now let's talk about why. Derivative of cosine is negative sine, so you end up with a negative times negative, which becomes positive and you're left with sine of x. Notice how we still need our constant of integration because there could have been a constant here, the derivative of which would be zero and would zero out when you take the derivative of it. Let's consider the cosine function, anti-derivative of cosine of x dx. Think to yourself, what can I take the derivative of and get cosine as my answer? Hopefully you've thought of the fact that if you take the derivative of sine, you know that to be cosine. Therefore the anti-derivative of cosine of x dx is sine of x plus c. You're going to notice as we go through here, you're not going to see yet anti-derivative of tangent and secant and cosecant and think about why. There's no really easy function that you know of from your derivative rules that when you take the derivative of it, you get tangent, nor can you get just plain secant or cosecant. But think again of your derivative rules. If I wanted the anti-derivative of secant squared of x dx, think what do you take the derivative of to get secant squared as your answer? Hopefully you've remembered that the derivative of tangent is secant squared. Therefore the anti-derivative of secant squared dx is tangent of x plus c. This is why you're saying you need to know your derivative rules really well and if you do, hopefully these make a lot of sense. What about the anti-derivative of cosecant squared of x dx? Think of your derivative rules. What can you take the derivative of and get cosecant squared of x as your answer? That would be the negative cotangent of x plus c. Now think about why derivative of cotangent is negative cosecant squared. Similar to our sine rule, the first one we looked at, you end up with a negative times negative and that of course turns positive and you end up simply with cosecant squared of x. What about the anti-derivative of secant of x tangent of x dx? Think again of your derivative rules. What did we take the derivative of for which we obtained secant of x tangent of x as our answer? That would be secant of x. We know derivative of secant of x is secant tangent. Notice how these are just, as I like to call them, the trig derivatives backwards. You're simply going backwards from the derivatives you already know to be true. There was one more derivative rule for trig functions you knew. What about the anti-derivative of cosecant of x cotangent of x dx? Think what did we take the derivative of and we got cosecant cotangent as the answer? That would be negative cosecant of x plus c. This was another one in which derivative of cosecant of x was negative cosecant cotangent, so the negative times negative turns positive and that's why you don't have a negative here in your integrand. Perhaps you remember back when we did derivatives, we had a little way of knowing which of our derivative answers were negative and we said that if what you're taking the derivative of began with the letter c, then your answer was negative. Well remember, anti-derivatives are just going backwards. So here we have, if your anti-derivative answer begins with the letter c, they are the ones that are negative. Let's consider now the inverse trig functions. There are only three you are required to know. If we have the anti-derivative of a function that looks like 1 over 1 plus x squared dx, that anti-derivative is arc tangent of x and if you think of your arc tangent derivative rule, it was 1 over 1 plus x squared. Doing anti-derivatives like this, it's really a matter of recognizing the pattern of the integrand. So this is the first of the three inverse trig function rules you need to know. Second one is for arc sine, inverse sine. If you have the anti-derivative of 1 over big square root 1 minus x squared dx, that anti-derivative is arc sine of x plus c. The last one is for arc cosine, which is simply the same as arc sine, but negative, think of the little hint I gave you. If the answer to your anti-derivative rule begins with c, then it's negative. Well think of this as a cosine negative 1 of x, then it begins with c. Otherwise you'll notice they are identical to each other. Once again, when you're working with anti-derivatives that are going to involve trigonometric functions, you often will have to rewrite that which is being anti-differentiated, the integrand, typically by using our Pythagorean identities. For example, if you wanted to do anti-derivative tangent squared of x dx, we'll think of all the derivative rules you know for trig functions, none of them deal with tangent squared. So this is a case in which you'd have to look at this and recognize the fact that you need to turn tangent squared into something for which you have an anti-derivative rule. Remember the Pythagorean identity 1 plus tangent squared equals secant squared. Therefore tangent squared is secant squared of x minus 1, and we have an anti-derivative rule for secant squared of x. So that's the trick. And sometimes the difficulty when dealing with the trig functions, you need to be able to look at it and recognize that you do need to rewrite it by using a Pythagorean identity or one of the other trig identities.