Evaluate the Indefinite integral: (Us Iot tho conatant intearatior. Ramembrr utuabeoluto Valueramuert appropraleraF(k)Tulorial...

Find the general indefinite integral.
$$\int\left(u^{6}-2 u^{5}-u^{3}+\frac{2}{7}\right) d u$$

They wanted to take integral on it. By component we have either than I get three. He's when goes on TV man, team, deity and I have done it by component. We're going to get negative one day into the negative. Wriggling Teoh London sign. We'll take it. Verify that because I never not one thing. And then the last one on the integration seeing we'll have tea sign of It's a chance. I see you know, uh, Santee, But being less because I have seen what's the event mix? We're going to take the product or offer. Then the 1st 1 I'll give us t cause I empty And also a sign of C term goes negative science something. It's at the arbitrary constancy.

We want to evaluate that definite interval he to the secret pie Tea time secret pie Tee Times 119 with perspective Now, in this chapter, we know that if we integrate e to the X the X we should get beat the ex off, see? So if we can somehow make some substitution to make this look like sex, this would become a trivial problem and notice if we let the power here of e u So u is equal to seeking high tea. And we take the derivative of this function with prospective tea. So we get the new is equal to Well, the driven of Sikkim is going to be seeking high tea Herb derivative seeking is seeking Handwara. But since we're using how herb chain role here, we would have that. And then we would also need to take the derivative of our inside function, which is high tea. And the derivative of that is just pie. That's why now let's divide over by pi. So we get you over. Pi is equal to secret pie t Oh, and I forgot my beauty on the outside. Here you get high tea handed, high tea and DT and you might notice that what we have in the right hand side here is also what we have being multiplied I e So we can read about this in your role as the to the U times. Do you divided by pie so we can factor that pie out front? So we get one over pi times e to the hugest, using the identity on the top right of the board, Plus are constant and then we can go back in and plug in what you was and you iss seek it. Hi. So the definite control or the indefinite integral of this is one of the pie times using a secret party.

You want to evaluate this indefinite integral Here Now, in the chapter, they tell us that if we integrate into the you, that should give us heat to the U policy. So maybe if we let you be seeking Pi ti we will get stuff to Counsellor actually, assuming a little So seek it Hi teeth Now the derivative of this. So our differential is going to be so first the derivative of seeking ISS seek int tangent But then we need to do change rule Take that route on the inside here which is going to give us a pie. Then we'd have pie DT. And now that's basically what we have right here. If we were to multiply this by pie and then divide by pi because pi times one replies just one So we didn't change the expression All So we're going to replace all of this here D'you started have one over pi and to roll you to the u do you which then integrates too. Just eat the u Well, see And then we want to take you and plug it in, which gives us one over pi e to the seek it Hi t plus c So this would be our solution to this definite in death

It's another integral of geometric function. So. Well, by the way, the question I have in front of me actually has X over six years, but I'm assuming that's just a type of so and it should be pi over six. That it would make sense for it to be X. So anyway, so once again, your textbook has a table of common chicken and metric anti derivatives and it tells us that the anti derivative of casa que katsu is actually just negative. Co sick you if I'm correct and I am. So that is just Taken from Pi over 6 to Pi over four. So that is negative. Co sick, you negative Kosek In the Point Pi over four minus negative. Co sick by over six. And remember that Cossack is just one over sine. So that's one over sine by over four minus negative one over sine pi over six. Mhm. And remember The sine of pi over two is just one over square too. And then over here, the sign of a sudden Pi over six is just a half, so that is just square too minus negative too, and that works out, and that's just ends up being two minus square root two. And there we go.