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1 Using None suitable 41 1 substitution; the integral 0 2 Il dr is given by:...

Question

1 Using None suitable 41 1 substitution; the integral 0 2 Il dr is given by:

1 Using None suitable 41 1 substitution; the integral 0 2 Il dr is given by:



Answers

In Problems 41 and $42,$ use a substitution to establish the given result. Assume $x>0.$ $$ \int_{1}^{\sqrt{x}} \frac{1}{t} d t=\frac{1}{2} \int_{1}^{x} \frac{1}{t} d t $$

For this problem we need to use a substitution so that the integral from 12 X squared of one over T. D. T. Is equal to twice the integral from 12 X. Of one over T. D. T. We begin by applying substitution to the left hand side of this equation. Now the integral from 12 X squared of one over T. D. T. If we apply a substitution and you want to let U equal to let's say the Square 50. And so to get an expression for T. In terms of you, we will square both sides of this equation and we will get you squared equal to T. And then to get an expression for DT we will take the differential of this equation. We get two times you do you. This is equal to D. T. Next you want to change our bounds from T to you. Now if T is equal to one, then we have You equal to the square root of one that is equal to one and if is equal to X squared, then we have U equal to the square it of X square. This is equal to X. And so by substitution we have The integral from 12 x of one over U squared times DT. Which has two times you do you. And this will give us two times the integral from 12 x Of one over you. D. You. And since you is just an arbitrary notation or variable, then we can say that the integral from 12 X squared Of one over TDT. This is equal to Two times the integral from 12 x of one over TDT.

Again this question. We have to solve integration. X divided by two X plus one whole cube DX and the limited 0 to 1. Okay, so we have to solve this. So we will do the substitution here, and we will substitute u equals to two X plus one. Okay, it will give us Do you equals two. Two d X. Or we can say the value of DX will be d'you divided by two. Okay. And from here, the value of X will be value of actual B U minus two, divided by two. Okay. And now we will apply substitution here. Okay, So, before a blank substitution, we will have to get the new limits here. Okay, So the limit, all the limiter is four X is zero. That is lower limit, and we have taken u equals to two x plus one, and we put X equals to zero. It will be u equals to one. That is our new lower limit. Okay. And Mexico's 21 is old, a parliament and we have taken u equals to two x plus one. And it will be when we put X equals to one. It will be your request to three. And this is new apart limit. Okay, so our question look like integration. It will one and a parliament is three. Okay. And in the numerator there is X. It will be u minus one, divided by two in the denominator two x plus one whole cube, that is you Cube and DX will be. Do you divide it by two? Okay, so we have to solve this, and it will be integration. 123 Okay, it will be u minus one, divided by two you cube dot Do you divided by two? Or we can say I can see integration. 1231 by 211 by 21 by four Outside the integration, it will be u minus one. Divided by you, Cube. And do you okay? And now it will be won by 4123 And we will divide you Cuban to U N. Minus one. It will be you divided by you, Cube. Okay. Minus one. Divided by you, Cube. And do you okay? And now it will be one divided by four and integration. 123 You divided by YouTube. That is one upon us choir or we can say you raised to the power minus two. It will be you raised to the power minus three. And do you okay? And now we will do the integration part here. So integration of us to the power minus two will be one of minus one up on you. Okay. And you raised to the power minus. You will be you raised to the power minus two. Divided by minus two. Okay. And the limit is one and a parliament history. Okay. And now it will be won by four. Okay. And it is minus one upon you and his hair. Minus minus. Plus, once upon to use square one and three. Okay. And now we apply the limits. It will be won by four. And now Parliament three, That is minus one by three. And plus one upon two and three square. That is nine. Okay. And here again, minus one by four. Common and minus one upon you. That is minus one by one. That's going up onto one is square. That is one. Okay, now it will be one by four. Okay. And this will be minus one by three. This will be one plus one by 18. And this will be minus minus. Plus one by one. That is one. Okay. And here, negative. One by two. We have taken one by four, again common. And one by four. Okay, it will be. We have taken LCD 18 here. Okay? And this will be minus six. Okay, then it will be plus one. And it will be plus 18 and minus nine. Or we can say it will be won by four. Multiply four by 18. Okay, that is one by 18. And one by 18 is our final answer. Thank you.

Again this question. We have to solve integration. D x divided by X access to the power four plus one. Okay, so we will rewrite our question as integration D d x divided by x. Okay. And we will take access to the power four comin out of this. It will be access to the power 41 plus access to the power minus four. Okay. Or we can say it will be d X divided by X raised to the power 51 plus X raised to the power minus four. Okay. And now we will take this portion has substitution. You okay? You will be one plus X raised to the power minus four. Or we can say do you will be minus four X raised to the power minus five dx OK, or it will be. Do you divided by four and minus will be one upon express to the power five dx. Okay. And this is presenting our portion. So it will be this d x, divided by express to the power five will be minus d'you divided by four. Okay. And this will be you, so it will be even upon you, okay? Or we can say our question will be minus one upon four. Integration do you divided by you and its integration will be minus one upon four. An integration of and by you do you will be learn You plus see Okay. And now we will back. Substitute the value of you That is one plus X raised to the power minus four and it will be minus one by four. Okay. And Ellen, one plus access to the power minus four plus C or we can say it will be minus one by four. Ellen, it will be one plus one upon express to the power four plus c. Okay. And to get it simple, it will be minus one by four. Okay, island, it will be expressed to the power four plus one, divided by express to the power four plus c. Okay. And we will have We have to eliminate this negative sign. So this will be won by four Ln It will go to the universe of the power of X. Okay. And it will be upon We will be be upon a okay, and it will be access to the power four divided by access to the power four plus one plus. See? Okay. And there is no need for the absolute value here because express to the powerful is a positive value. And this will be our final answer. Thank you.

Again Discussion. We have to solve integration. D x divided by X square root for X square plus one. Okay, so in discussion we will substitute X equals +21 by 2. 10. Okay, so why we have taken one by 2. 20 to because one by two and square will come one by four and it will be cut by four. Okay, that's why we have taken exception to one by 2. 20. And it will use us. The X will be won by 26 square to to date. Okay. And now we will put the values when the numerator DX will be won by 26 square to to to divided by X is here that will be half, 10 to to and in the square root. There will be four and one by 2. 20 to holy square plus one. Okay, now it will be It will be integration. Half will be cut by half. Okay. And six square data. Due to time the numerator and the denominator. It will be 10 to okay and square root. This will be one by two. Holy square will be won by four. This will be cut by this four and it will be 10 squared plus one only. Okay? Or we can say it will be integration. Six square to duty to divided by. It will be 10 to Okay. And this will be sector. Okay. And now it will be one sector will be cut into this and it will be integration sector to did it divided by 10 data. Okay. Or we can say sector will be one upon cost. Rita divided by attended. I will be sign upon course. Okay, and data to okay. And this can be written as it will be because to that it will be when a bomb goes to the okay and divided. A sign of phone calls can be written discourse upon sign do okay, or we can say it will be dictator divided by scientists. Now it will be integration core sector to Cosette to to do and the integration. Of course, it will be Ln core sector to so the core sector to minus Got it up plus c. Okay. And now we have to find the values of course sector to and quarter to, as we have put tend to take us to okay. X equals to one way to tend to do. Okay, so now we have to find the values, of course, sector time, court data. And we have taken X equals 21 by 2, 10 to or we can say 10 total will be two x. Okay, then surely court. It will be won by two weeks. So this is the value of quarter to okay. And now for the value of course sector to we will draw and let's take. This is Tita and tended to is two weeks. Then we can say the 14 years will be too works and the base will be one. Okay, okay. Sorry. Perpendicular is two X and bases one, then hypertension will be rules for X square plus one. Okay, So connected to will be Hi pate Gnaeus divided by I put any divided by perpendicular and it will be for secretary will be mhm root for excess choir plus one divided by perpendicular is two x. Okay, So this is the value of course, dictator. And this is the value of porchetta. We will put it here for spectator will be wrote four X square plus one divided by two x minus quarter to will be won by two X. Okay. And plus C. And this will be our final answer. Thank you.


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