5

6_ Find & for Y =sin? (Sx) dx...

Question

6_ Find & for Y =sin? (Sx) dx

6_ Find & for Y =sin? (Sx) dx



Answers

Use Equation 6 to find $ dy/dx $.
$ e^y \sin x = x + xy $

Equation six says that to you my DX can be written as the following. Now, this is assuming that the original equation is of the form capital F equals zero. So in our case, we can go ahead and combine everything to the left hand side. So then we could just go ahead and call this our capital f Now we'LL use thie equation over here. This is Equation six from the textbook. So first, let's go ahead and find this numerator here. The rivet, um of our new expression capital if with respect to X So here we have X So first differentiate co sign and then using the chain rule. Well, excuse me. And for that my my ex in here and then using the chain room well supplied by the inside with respect to X and that'LL give us why and then both of these terms the negative one and minus sign Why those air constant with respect to X So the derivatives of zero Here's another derivative So co sign you get a negative sign this time will supply by the derivative of X y What respect? So why giving you x and then the ones of constant that goes to zero and then negative sign Why becomes negative coastline way? So let's put all this together And here you can also go ahead and just cancel out that double negative on the top. So if you do that actually, let's go ahead and cancel these negatives here we'LL get negative. Why sine x y And then in the bottom those both turn into a plus Signs. Yeah. No. And this right here. This is our final answer.

This time they want us to find that relative of the function wise equal to eat. Six x Santa. Let's do you predict role. Take the derivative of the first time really, with 2nd 1 alone. Then leave the first term alone and take the derivative of the second term. And this is ah, derivative six and Iraq's plus goes on your ex.

To integrate the integral of sine races. The sixth power of X. Dx. We would first write this into the integral of the cube of sine squared X. D X. And then we would use the half angle identity for sine squared X. That's just 1- Co sign of two x over two. And so we would get the integral of the cube of one minus co sign F two X over two. And then the X. and this simplifies to 1/8 integral of 1 -3 co sign of two. X Plus three co sine squared of two, X minus co sign cube two X. And then dx. And then from here we integrate term by term and we have 1/8 Integral of one d. ex or DX 3/8 and a roll of co sign of two X. Dx plus 3/8. And the role of co sine squared of two X. Dx. And then lastly we have 1/8 integral of co sign to the third power of two X. The X. Now, for the first in a girl we have 1/8 integral of dx. This is the same as 1/8 X. And then for the second one we have -3/8 integral of CO sign up two X. Dx. This is just negative 3/8 Times sine of two x over to Which is the same as negative 3/16 Times Sine of two X. And then for the third one we have 3/8 times the integral of cosine squared of two X. Dx which is the same as 3/8 integral of In this case we want to use the half angle identity for co sine squared of X. That will be one Plus co sign of two times 2 x 74 x over to. And then dx. And this will give us 3/16 integral of one plus co sign of forex which is equal to 3/16 times you have X. Plus we have sign of four x over four Which is equal to 3/16 x Plus 3/64 Sign of forex. And for the last one we have negative 1/8 of the cube of co sign F two X. Which we can rewrite into negative 1/8 integral of cosine squared up to X times co sign up two X. Dx. And then from here I wanna change go sine squared into its identity one minus sine squared of two X. And so we have the integral of one minus sine squared of two X times. Go sign of two X dx. And in here we want to use substitution. We let U equal to sign up to X. So that the U. Is just co sign of two X times two dx. Or that 1/2 d'You is Co sign of two x. DX. And so we get negative 1/8 and a girl of one minus U squared I was 1/2. You. This is equal to -1/16 and a girl of one minus U squared do you? Which is equal to negative 1/16 times U minus U. To the third power over three. And since you signed up to X, this is just negative 1/16 times sine of two X minus sign of two X to the third power over three. And now adding all of the terms we have The integral of sine to the 6th of XDX. This is equal to 1/8 x -3/16 sign of two x Plus 3/16 x Plus, we have 3/64 signs of four x -1/16. These times Sign of two x minus sign cube of two X Over three and then plus C. And this simplifies further into 5/16 x -1 over. for sign of two x Plus 3/64 sign of forex Plus, we have 1/48 sign cube of two X and then plus C. And so this is our indefinite integral.

We wonder Integrate sign six packs. There are compliment. That's what This question we will use substitution method. So let's you it goes to six X when a different shit you I would differentiate six x So I get six and let me d x so our head 1/6 signs six x and I put a six year DX So it will be signed you since six x is you and 60 x would be Do you now We all know where integrate sign will get minus school sign plus c no, we cannot leave our answer in you. We have to change it back to x so would be minus 16 Cool sign six x last see?


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