Evaluate the integrale-(s+4)cos(2t)dt(s+4)2+4(s+4)(52+4)C)D)E)e-4s+1 (5+4)...

Evaluate the integral. $$ \int e^{3 x} \cos 2 x d x $$

Because I have an E. And a trig function. I recognized this as one of those repeating right? And the girl so I'm gonna call it. I I'm gonna let you be eat to the three X. Dvb. Co sign for X. Dx. So do you is three E. To the three X. Dx envy is if you let for XP you then do us four D. X. So I need a four and 1/4. So that's 1/4 for x. So now I is UV 1/4 each. The three X. Signed for X. Minus the integral VD you. So three force. Each of the three X. signed for X. D. X. Whoops. All right so I'm gonna let you be each of the 3X. Dvb signed for X. Dx. So then do you is three eat the three X. Dx. And V. Is if that's you then do us four D. X. So I need a fork and I 1/4. So that's -1 4th coastline for X. So now I have I equals 1/4. Eat to the three X. Sign for X. What's that -3 Force. That's that square bracket U. V -1 4th. Eat the three X. Coastline forex minus plus minus. That makes plus in a row V. D. You so 34 eat to the three groups in a role. Each of the three X. Coastline four X. Dx. Okay but that is what we're looking for that. I so now I have I equals 1/4. Each of the three X. Signed for X. Plus 3/16. Eat the X. Each of the three X coastline four X -9/16 i. Okay so I'm going to add 9/16 ii to both sides. Yeah. Okay. 16 16 plus 9/16. That's 25 16 25 16. I well it's 1/4 to eat to the three X. Sign for X plus 3/16. Each of the three X. Co sign for X. So I'm going to multiply everything by 16/25 So I get I equals 4 25th. Eat the three X. Sign for X Plus 3 25th. Okay. Each of the three X co sign for X plus seat.

So for the following in a girl. Well, here we have a constant integral of a constant with respect to X is always just si x. And then I guess we are constant after that was a plus de since every UC and I guess the quick way to verify this is, well, it's just using the power rule. So here, if you want to integrate just some constant see, you could pull out the sea, and then you have inaugural one d X, and then you could write that as CNN roll X to the zero and then use the power rule. So then you just couldn't see X, and then you had your constant at the end. So due to the fact East where it is just a constant So I just get e squared eggs and then add my constant of integration, see? And that's my final answer.

So we have integral off e to the X Times. Go seek it three e to the X plus four DX. So let's do some substitution here. So I'm gonna take you as three e to the X plus four. So do you. Over DX is three ah three e to the X So d x is do you over three e to the X Now let's start substituting now Give us e to the X Times Co seek it you times do you over three e to the X So this here he to the X's cancel We can take one out of three out so we'll end up with 1/3 Cosi, can't you Do you now the integral of co ck and is standard It is minus Ln so that's times minus. Ln off um Cosi can you plus co tangent you remember you is three to the three e to the X plus four. So our final answer is that's 1/3 times. Let's just take the negative out here minus 1/3 Ln of go seek and three e to the X plus four plus co tangent three e to the X plus four. So this here is the final answer

We won't integrate he to the power tree monies to X. There are a couple minutes for this question. We were used substitution method. Let U equals two tree minus two x. When I differentiate you, I'll be differentiating my right side. So I have minus two the X. So I really need a minus two here. So what I would do is output minus half here. So I'll have a minus two year the X and either about tree minus two x. So now I can replace these with you. And this week do you? So I have minus half you. You when we integrate e how you would just get eat about you. Let's see where sees a constant replace back there you with three minus two x So I get this. Let's see.