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K) i) g 3 J 8 7 tan (x-2)3/2 1112X dx dx dx...

Question

K) i) g 3 J 8 7 tan (x-2)3/2 1112X dx dx dx

k) i) g 3 J 8 7 tan (x-2)3/2 1112X dx dx dx



Answers

$\int \operatorname{tab}^{6} x \sec ^{2} x d x=$ (A) $\frac{\tan ^{7} x}{7}+C$ (B) $\frac{\tan ^{7} x}{7}+\frac{\sec ^{3} x}{3}+C$ (C) $\frac{\tan ^{7} x \sec ^{3} x}{21}+C$ (D) $7 \tan ^{7} x+C$

We have the integra X. Keep. So I equal to s keep plus two X divided by X. To the power for Plus four x squared Plus three d. x. And we can rewrite this as the Ziggo. You have s key plus two x divided by I have X squared last three. Then X squared plus one. The eggs. So we can write the Russia fracturing the composition for this. So then it's in place. That's our S keep plus two eggs divided by X. Work X squared air squared plus. To me Have a sweat platz one two vehicle to A X plus B. Divided by X squared. Last three. Plaid C. X. Blas de divided by X squared bloods one. And this is equal to this will be E. X plus B. Ex Blaise B. They have X squared last one Bless C. X plus D. You have X squared. Bless three. You write it by, you have X squared last three. You have X squared platz one. So you quit an emeritus. So I have this in place. I have eggs, Q. Plus two X. To be called to A X. Plus. He you have S squared plus one plus C. X plus D. I have X squared plus three. So you saw for values of a. b. c. and D. So when you saw you realize that this implies that I. E. It's equal to one divided by two game. Mm. He is equal to zero. The M. T would be equal to zoo. So see to have C. To be equal to one divided by two. The M. T. to be equal to zero as well. So now we substitute in the coefficients and we integrate. So we have so this implies that we have the integra From seven. So when you have X cubed plus two X divided by X. To the power for Plus four x grade. Last three the S to be equal to the eggs. Iga. Thank you girl. The integral from 0 to Juan. You have one divided by two x divided by X squared plus three And plus one divided by two eggs divided by X squared plus one D. X. Sudan Using the substitution may sit here you realize that this is going to be equal to one. So we have one invited by soon. The secret conservative one of X divided by X squared Last three. So from here from here this is where we have plus eggs divided by X squared plus one. The eggs This will be equal to one divided by two. We have one divided by two lane X squared plus three Plus one divided by two blame X squared flats one They eat ever from 0 to 1. So this would be equal to The constant out. So you have one divided by four. You have lain four less lane 2- the other path which is lane lane three last lane one and this will be equal to This will give us one divided by four. Len. So then one divided by falling eats minus Lane 3 -74001. And this is equal to This is equal to one divided by four lane. It's -63 And this is equal to one divided by four Lane. It's divided by three as the final answer.

We have the integra. So let's give I I see be equal to the Utica. X cube divided by esky bloods one the X. So I can write S keep here. So s keep divided by sq platz one. We can write this as one mine. It's one divided by sq blood one. So let's rise plus your fracturing for this. So the partial fraction. Let's see pasha fraction for that is going to be you have one divided by S. Q plus one. To be equal to have E. Divided by X plus one plus B. X. Let's see to give you X square minus X plus one. So this in place. That's from here. You have one divided by sq platz one C. B. Equal to a divided by sq be divided by X. Last one plus B. X plus C. Divided by X squared minus X plus one. So this implies that you have 12 vehicle to a. Which is X squared minus X plus one. Bless mm X plus C be express C. You have X plus one. So this implies that. So from here it's implies I. E. It's equal to one divided by three. Yeah, It's equal to one divided by three. Then see it's equal to negative suit divided by T. V. So then from here I. So from my eye which is an integer is cube divided by Ask you plus one. The eggs is going to be one this 1. So you have the integra. You have one minus this. What's this? E. is one divided by three. So that'll give us one divided by Jimmy X plus one. Then then despite B. And C. You have plus B. is one divided by T. V. one divided by Theories. X plus C. SES- or -2 divided by three. You fight it by X squared minus X plus one the X. So this thing is going to be equal to plain cigarette one the X one the eggs -1 divided by three day. In Syria 1 to X plus one. The X. Plan ST zebra. You have been Sequera X minus two divided by three. Expert -3 x plus three the X. So this is going to be equal to X. My next one divided by three lane X plus one plus thing. Seeger X minus two. You have three X squared minus three x plus three D. S. So we need to solve this parts. So the strategy to solve for the remaining insignia is to have the derivative of its denominator. So so from here we are saying have the day we receive of its denominator. So then this implies that you have X. Grade minus X plus one. They eventually walked dads To be equal to two eggs. My next one this in the memory to the new Reza in the room so we can define the and see their effective for that's a good name. This implies that our I is going to be called to Our eggs. My next one divided by three Plain x plus one. Let's one divided by six. We helped insignia. two x -1 divided by you have X squared minus X plus one. The eggs. My next one divided by two. The n sinker one divided by X minus one divided by two squared Last three. invited by four. The eggs. And this would be equal to. So then I integral will be equal to s -1 divided by three lane X plus one. Blood's one divided by six Lane. You have X squared minus X plus one minus well One divided by square roots of three. Time to invest Of. You have two eggs. My next one Divided by Square roots of three. Let's see as the final. And so. Uh huh.

Factor nine out of the denominator to write as one nights when we pull it out. Then we flip it since the tonight over, when it's one night times DX on the top and one plus x over three squared. Now use U substitution you is X over three. Therefore, d'you is 1/3 dx and three d U equals DX. So now, given this, we have 1/3 do you over one plus u squared. And this actually looks a lot like the inverse tan function. So you're gonna write 1/3 10 to the minus one, which means in verse 10 of you plus C and the final step is you just substitute back in your you and then you have your answer.

We want to find the integration to this function. First of all, we recognize that is an improper function, because the power in the numerator is greater than the power in the denominator here. So let's do long division extra for four divided by X minus one. So we have X Cuba little happiness, so we'll get executed. So plus now X square. So we have excuse yet Do we have X square again? So we had plus x x square your ex here. Now remember, it's one. So first we're putting down this fraction into execute plus X square plus X plus one plus one over X minus one. Yes, so excuse when we integrate with X for over four x squared when we integrate the X cubed over three extreme ingredient X squared over two one women integrate. We will have X and one mobile X minus one. When the great we get long Model X minus one and finally plus C


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