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/4 sec x tan? x da14. Evaluate...

Question

/4 sec x tan? x da14. Evaluate

/4 sec x tan? x da 14. Evaluate



Answers

Evaluate the integral. $$\int \tan 4 x \sec ^{4} 4 x d x$$

Find the angle. Whose tangent is the changing of power before? What that means is we're finding the tan in verse of the tangent, the power Before Now. First, let's find the change in a powerful power forwards in Quadrant one. This would have a 45 degree reference angle. So here's the relationship between the sides. And so the tangent would be won over once the opposite over adjacent. So this is really the ankle whose tangent is one? Well, guess what? The angle whose tension is one is power for our changin is defined in quadrants one and four. Since the tangent is positive, we know we're in quadrant one and so one written as a fractions 1/1. That angle is, uh, over four.

The human expression is Dan. Uh then in worse minus four point toe here, minus 4.2 is in range off 10 in Worse. Hence done. Well, you off then. Off then in WAAS minus 4.2 is equal to minus 4.2.

We have to find out where you are. They're in worse. Dad. Quietness three by the wife by four We know pendant is positive in third party s Yeah, minus three by the went by. Full is equal toe. Yeah, Why be widened by full s legal right Then it waas uh, dad by they went by full his equal Do Tita hence, when you is by, you want by full

Okay, we're still integrating powers of trig functions. Tangents and sequence are a little more complicated than signs and co signs on a tangent and seeking power problem. I can either choose you to be tangent or I can choose you to be seeking and but there are different results each way and they'll all look different but they're all equivalent values. They're all equivalent expressions. So um I just get to choose whether I'm going to deal with sequence or deal with tangents. Remember that the derivative of tangent is second squared and the derivative of seeking it is seek it tangent. So my first job is to collect up some derivatives. I can either choose second squared dx to be a derivative or I can choose a second plus one tangent to be a derivative. And then the rest of the tangents would switch to seek its So I got to switch everybody two seconds or switch everybody to tangents because tan squared X plus one is second squared X. A function in terms of tan squared is just a vertical shift of a second squared. And they can again, I can get two different answers completely equivalent. So what I'm gonna do is I'm going to work this using U. Equals seek it of four X. Because I think the books answer was in terms of seeking. So I want you to see the equivalent. I want you to see the answer that you feel comfortable with. But I could have just as easily chosen you as you to be tangent and switched everybody else over. Alright. You is seeking for X. So D. U. Is seeking for X. Tangent for X times the derivative of what's inside. For terms of dx So out of this integral, I'm gonna take one of the tangents and one of the secrets. So I have 10 squared for x Times seeking cubed of four x Times tangent for X. Seek it for X. D. X. So I have 1234 sequence And I have 123 tangents and I still have a DX 1/4 of the D. U. Is the same as this part back here. I have a U. Cubed tangent squared isn't anything yet but I know 10 squared plus one is second squared so tan squared is second squared minus one. And that's gonna be you squared -1 now that I'm writing things in terms of use So I'm going to integrate 1 4th U. to the 5th -U. to the third do you? That's 1/4 You to the 6th over six -U. to the 4th. We're seeking to the fourth. Four X. Over four plus C. That is my final answer. A little bit ugly. Thanks for listening though. Talk to you later.


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