5

Pt | ,'' "10ttk'vsluste the detlmite imekrmi93 52_^Evaluate the imtewal4x2 0 dx x2+910. Evaluate the integral...

Question

Pt | ,'' "10ttk'vsluste the detlmite imekrmi93 52_^Evaluate the imtewal4x2 0 dx x2+910. Evaluate the integral

Pt | , '' " 10ttk 'vsluste the detlmite imekrmi 93 52_^ Evaluate the imtewal 4x2 0 dx x2+9 10. Evaluate the integral



Answers

Evaluate the integral.

$ \displaystyle \int \frac{4^x + 10^x}{2^x}\ dx $

Let's go. And this problem, we have the integral of three X squared minus 10 over X squared minus four X plus four Y Yeah, it's okay. So let's look at this so we could factor the bottom. However, since the new Warner has the same power as the denominator in terms of polynomial miles, we're gonna want to fix that first eso. One thing we could do is we can try to make the top look like the denominator. Or we could just use long division. So I'm going up for a long division. So we're gonna have three x So x squared minus four x Plus four is dividing into you three x squared plus zero x minus. Stan, I just put the zero AC. Syria's a placeholder. Just makes a little easier for myself. So we got almost by by threes like it's us three X squared minus 12 x plus 12. And when we subtract up, uh, that zero that's gonna be 12 acts, and that's gonna be negative. 22. Great. Okay, So where does that leave us? So, from here, we can write this as plus 12 x minus 22 over, um, X wearing my ass for X plus four. Right. So three, right that you're and we're gonna have Ah, three plus 12 X minus 20 is you divide by. I was fact us. We get X Plus two on a squared DX. Okay, so, uh, let's go ahead and two partial fractions. So on this. Right, So we have 12 x minus 20. Let's go ahead and factor to now as we leave it. So 12 x, I asked 22 over X plus two already squared. So this is repeated. So we're gonna write this as a over exploits to go. Three of my asked you. Yeah, you picked up my ass. Where? Yep. Okay. Looks good. All right, so this one here plus B over X minus two. Nice work. OK, so, uh, we could find being using the heavy side cover, but that's with X equals two on. We evaluate it. We get B is equal. Teoh 24 minus 22 which equals two. Okay, so then we have 12 x minus 22. We most by the denominator, will get equals a times X minus two plus. Do you most u plus U. Yep. Okay, so we get 12 X minus 24 equals a times X minus two A equals 12. Next. So do let's reread the inner grind again. So we have three plus 12 over Experienced you plus two over X squared minus two x minus you X minus two Bonnie Square DX. Right. We use U substitution here, and we can integrate these directly, so u equals X minus two U equals TX. So we at three get three X plus 12 Ellen absolute value of X minus two plus to enroll of one over you square you what? Some constant because we integrated. So we have three X plus 12. Ellen. Absolutely. You, uh, X minus you bus two times a one times you to the native one constant. So that's three X plus 12 Ellen, Absolute value X minus two um, minus two over. What was you view is explain this to you and that completes our problems

OK, so using long division and then factoring out the denominator, we can get the integral from 0 to 1 off X plus one D x, plus the integral from 1 0 to 1 over the three x minus four. Divided by X plus two explodes two times X minus three. So, um, if we saw this using partial fractions so this becomes a over X plus two plus B over X minus three, multiplying both sides by the least common denominator, we end up with a multiplied by X minus three plus be multiplied by X plus two. Setting X is equal to negative, too. We end up with a is equal to two and sending X is equal to three. We end up with B is equal to one, so therefore we can be right our this integral over here as the integral from 0 to 1 of X plus one the X plus the integral from 1 0 to 1 off to over X plus two plus one over X minus three DX. So integrating this we end up with X squared, divided by two plus acts plus two ln of the absolute value of X plus two plus Ellen of the absolute value of X minus three plus c. And then we're gonna evaluate all of this from 3 0 to 1. So playing in numbers again, up with three over two plus two ln off three over two plus ln off to over three.

Right. Uh, first, look at your denominator. This is a quadratic and form polynomial. We can factor this by treating everything just like we would a quadratic equation on instead of X, we're gonna have X squared. So this factors into X squared minus nine and X squared minus one. And hopefully you notice that those were hurt the difference of two squares, So those will factor some more, which would do so in just a minute. So that means we can decompose this fraction off one over X to the fourth minus 10 x squared plus nine into a bunch of smaller fractions. And two of our denominators of these smaller fractions will be the factors of the X squared minus nine X plus three and next minus three and two other fractions that came from the factors of the X squared minus long X minus one and X plus one. So each of these being one your factors, we would just have some sort of variable as each numerator. So a over X plus three be over X minus three. See over X minus one D over X plus one is the form I'm gonna use for this So now we multiply all this by the common denominator to get rid of all the fractions. So we end up with one on the left hand side. When we have to multiply all the other fractions, the denominator reduces, and we're left with the numerator times, the three other denominators. So you have your choice. You can take your individual denominator times the other to the square that it came from and foil that real quick. What I actually did is I did synthetic division. I took my original denominator back here the X to the fourth minus 10 X squared plus nine on my did synthetic division with my numerator, our service, the denominator of that fraction to figure out what was left. So then I knew what I was multiplying the numerator by. So, for example, I did synthetic division with X Plus three, and I found out that X minus three times the X minus. One of the X plus one gives me X Cube minus three X squared minus X plus three and all that's multiplied by a. So we end up with a X cubed minus three. A X squared mine's a X plus three A. When we get rid off the X Plus three and same just doing the same thing with the B and the sea and the D fractions gonna have each variable times that remaining polynomial again. I used the synthetic division to divide out that denominator and get the polynomial that was left over. So just to Brock, jog down what does work that would then be X cubed plus three b x squared minus B X minus three B and then plus the seas c x cubed plus C x squared minus nine c x minus nine c and then plus rd's, that's the X cubed minus D X squared minus nine d X plus nine Deke. So that is what we get. Only clear out all the fractions. Now we can equate the terms on each side and the coefficients of them to get our system of equations. And so all of our X cubed terms here have to add up to the X cubed terms. On the other side, well, there aren't any. So that means that a plus B plus C plus d equals zero and all of our X squared terms have to add up to the X squared terms on the other side, which is also zero. So that means that negative three a plus three b plus c minus d equals zero and all of our X terms have to add up to the X terms on the other side. So negative day minus B, my ass nine C minus 90 equals zero because there's no excess on the left hand side. And finally, all the constants have to add up to our constant on the other side. So three a minus three B minus nine C plus 90 equals one. So now take whatever method you want to solve this. I plugged that into a graphing calculator, using the matrix forms in the Matrix commands to solve this. There's you can do it by hand. I'm old enough that when I was back in school, we didn't have graphing calculators and we would have to have done this one all by hand. Consider yourselves lucky. So what we end up with is the following answer. Whatever method you're using, a his negative 1 48th b is positive. 1 48 see his negative 1/16 and D is positive 1 16 which means that our original integration problem can be rewritten as the following. We have the integral of negative 1 48th over X plus three plus 1 48th over X minus three minus 1/16 over X minus one plus 1/16 over X plus one Oh d X And now we can integrate each one of these individually pulling our numerator in front and using the one over you. Do you formula for any girls. And so we have negative 1 48th times the natural AWB of absolute value of X plus three plus 1 48th times the natural log of the apse with value of X minus three minus 1/16 times the natural log of the absolute value of X minus one plus 1/16 times the natural of the absolute value of X plus one. Plus that good old arbitrary constant that we have for putting around which I'm just gonna let that sit there for the moment. And now, uh oh. Of these coefficients here have a common factor of 1/16. So I'm going to factor out no. 1/16 to help make Cynthy simplifying of this with our logarithms rules a little bit nicer. So I'm going to have 1/16 times negative 1/3 time. My algorithm of exposed three plus 1/3 time my longer rhythm of X minus three minus the logarithms of the X minus one plus the algorithm of the X plus one. And now we can combine these together the addition becoming more application my coefficients that I have here becoming exponents on the arguments and my subtraction becoming division. So I have 1/16 times the natural algorithm of the absolute value of the X minus three with 1/3 power times the X pulse one. Those are my two positive ones, divided by the ex post three to the 1/3 power times the X minus one being my to subtracting my two negative terms plus my c And now, if you wanted to, that 1/16 could then become an exponents on the entire argument here if you wanted to, I'm not sure if it looks any better, but you could have your algorithm off the absolute value of X minus three to the 1/3 x plus one over the X plus three to the 1/3 x minus one and that absolute value to the 1/16 power apostasy

It could be the one over for the day, except that over X to the fourth minus 10 next last night we'll end up with once you identify the partial a fraction decomposition in this important in your factors. Yeah, let's see. Let's pull out the 1/16. You got a negative one over X minus one. What's the one over at plus one and then a minus? Wondered one over X Plus three on a plus. Wondered one over X minus me. Actually, let's make this 48. Well, just multiply these by me and we won't have to worry about these. Intimate because they were actually taking is an entro la uh, Let's see. 1/48 friends three over X plus one minus the over X minus one plus one over X minus E minus one over X. Clustering to the order a little bit, the X. So let's take that And the derivative. We'll have one other party times but seen three times. Let's bring all under the similar. We'll have a natural I am a civilian Metro lock properties or last collect all their anti derivatives. We'll have an X plus one cubed over an X minus. One cube. Let's see. Time was an ex minister t in the numerator from the north subtracting so next close to be in the denominated off. Listen, arbitrary constancy, all right?


Similar Solved Questions

5 answers
NOz NHHCOz Pdc heat 10% HCINOzNHz
NOz NHHCOz Pdc heat 10% HCI NOz NHz...
5 answers
Remark lellel: ll are all negative Q 1S nd AII leading principle miors of Q are positive < QiS n dexaupleV()=21" +451X, 3x" 6x,4,lel=2 - 0 Tule He lel-6> 0 Qt Sh) alp Jel =-24<0 slerx; IS nol Dd
Remark lellel: ll are all negative Q 1S nd AII leading principle miors of Q are positive < QiS n d exauple V()=21" +451X, 3x" 6x,4, lel=2 - 0 Tule He lel-6> 0 Qt Sh) alp Jel =-24<0 slerx; IS nol Dd...
5 answers
The area 1' 0 areas 2 Homework: J: pt of parallelogram ABCD C(8,1) parallelogram D(3,4) 11 whose .4 square units vertices are given below:4of 5 (3 complete) MA 227 1
The area 1' 0 areas 2 Homework: J: pt of parallelogram ABCD C(8,1) parallelogram D(3,4) 11 whose .4 square units vertices are given below: 4of 5 (3 complete) MA 227 1...
5 answers
A) Alul cleaves DNA at a GTAC sites. A bacterial chromosome is 2.S6x106 bp. What is the best estimate of the number of fragments that results from Alul cleavage?
a) Alul cleaves DNA at a GTAC sites. A bacterial chromosome is 2.S6x106 bp. What is the best estimate of the number of fragments that results from Alul cleavage?...
5 answers
Mg1 7 that Sey what is the 3 Ahawk horizontal direction 1 Yeraviationai has alr magnitude = on 0f760 8 ? 7 the 12kg { W force
mg 1 7 that Sey what is the 3 Ahawk horizontal direction 1 Yeraviationai has alr magnitude = on 0f760 8 ? 7 the 12kg { W force...
5 answers
Ind the average value over the given interval 19) y=x2 3x + 6; [0, 8]
ind the average value over the given interval 19) y=x2 3x + 6; [0, 8]...
5 answers
Fcx) = e*? Ihal i5 the vale of f"1)
Fcx) = e*? Ihal i5 the vale of f"1)...
5 answers
Use the middle-square method to generatea. 10 random numbers using $x_{0}=1009$.b. 20 random numbers using $x_{0}=653217$.c. 15 random numbers using $x_{0}=3043$.d. Comment about the results of each sequence. Was there cycling? Did each sequence degenerate rapidly?
Use the middle-square method to generate a. 10 random numbers using $x_{0}=1009$. b. 20 random numbers using $x_{0}=653217$. c. 15 random numbers using $x_{0}=3043$. d. Comment about the results of each sequence. Was there cycling? Did each sequence degenerate rapidly?...
5 answers
Use the graph below to answer question 1 J00 9u NaNQ 3 80 0 70 CaCl 9 60 [b(NOsh 9 1 50 KC 1 40 NC 30 0 KCIO; 20 10 Cex(SO4ha 10 20 30 40 150 6 70 Xo Y I Temperalure (C)How many grams of KNO3 must be added to 100 g of water to form a saturated solution at 20o C? Enter your answer in the box below. Do not include a unit with your answer:
Use the graph below to answer question 1 J00 9u NaNQ 3 80 0 70 CaCl 9 60 [b(NOsh 9 1 50 KC 1 40 NC 30 0 KCIO; 20 10 Cex(SO4ha 10 20 30 40 150 6 70 Xo Y I Temperalure (C) How many grams of KNO3 must be added to 100 g of water to form a saturated solution at 20o C? Enter your answer in the box below. ...
5 answers
Find the LCM of each set of polynomials.$9 x^{3}, 5 x y^{2}, 15 x^{2} y^{3}$
Find the LCM of each set of polynomials. $9 x^{3}, 5 x y^{2}, 15 x^{2} y^{3}$...
5 answers
What is meant when a reaction is described as “having reachedequilibrium”? What does this statement mean regarding the forwardand reverse reaction rates? What does this statement mean regardingthe amounts or concentrations of the reactants and theproducts?
What is meant when a reaction is described as “having reached equilibrium”? What does this statement mean regarding the forward and reverse reaction rates? What does this statement mean regarding the amounts or concentrations of the reactants and the products?...
5 answers
Compounds Lead II nitrateMagnesium Iron ITI sulfate chlorideCopper sulfoteIron III chloridePotassium bromideSodium hydroxideSodium CarborateSodium Phasphate
Compounds Lead II nitrate Magnesium Iron ITI sulfate chloride Copper sulfote Iron III chloride Potassium bromide Sodium hydroxide Sodium Carborate Sodium Phasphate...
5 answers
To find the score associated with the highest 1% of normal distribution; recognize that the area t0 the left of thls score IsClick the answer you think is right2.33No IdeFa Bo
To find the score associated with the highest 1% of normal distribution; recognize that the area t0 the left of thls score Is Click the answer you think is right 2.33 No Ide Fa Bo...
5 answers
Which of the three series () L- E nnn) n =2 absolutely convergent? series () and (ii) series (i) and (ii)2 (~Ip - 13 (i) 2 (in-1 #@ n- n-1 nvnseries (noneseries (UJland
Which of the three series () L- E nnn) n =2 absolutely convergent? series () and (ii) series (i) and (ii) 2 (~Ip - 13 (i) 2 (in-1 #@ n- n-1 nvn series ( none series (UJland...
5 answers
Complete the table below:Complete Atom Mass Symbol or Ion number INAtomic numberNumber of Number of Number ofl protons Incutrons electrons1802- 107 A7X"1074710. Complete the table below Compound name Chemical (IUPAC) formulaTonicMMolecular or AcidicPotassium dichromateCuoHCIO4Sulfur hexafluoride Cobalt IIl carbonate
Complete the table below: Complete Atom Mass Symbol or Ion number IN Atomic number Number of Number of Number ofl protons Incutrons electrons 1802- 107 A7X" 107 47 10. Complete the table below Compound name Chemical (IUPAC) formula TonicMMolecular or Acidic Potassium dichromate Cuo HCIO4 Sulfu...
5 answers
Apply Newton's Method to f(x) and initial guess Xo to calculate X1, X2, X3_ f(x) = 1 1Ox sin(x), Xo =7(Give your answers to six decimal places )X]X2 ~X3
Apply Newton's Method to f(x) and initial guess Xo to calculate X1, X2, X3_ f(x) = 1 1Ox sin(x), Xo =7 (Give your answers to six decimal places ) X] X2 ~ X3...
5 answers
Chpose 11029.98
Chpose 110 2 9.9 8...

-- 0.021344--