5

To evaluate the integral Intlty dx by Integration by Parts, the convenient choice is (44)Oy=( + *)", du = Inxdx Oy=(+ X-? dv = Inxdx Int!+) dv = (1 +x)-"d...

Question

To evaluate the integral Intlty dx by Integration by Parts, the convenient choice is (44)Oy=( + *)", du = Inxdx Oy=(+ X-? dv = Inxdx Int!+) dv = (1 +x)-"dx 0 In(l + x) dv =( + *)"dr Ou=In( +x)du = ( +x)dx

To evaluate the integral Intlty dx by Integration by Parts, the convenient choice is (44) Oy=( + *)", du = Inxdx Oy=(+ X-? dv = Inxdx Int!+) dv = (1 +x)-"dx 0 In(l + x) dv =( + *)"dr Ou=In( +x)du = ( +x)dx



Answers

Use integration by parts to evaluate the given integral. $$ \int x^{4} \sin 2 x d x $$

Let's evaluate the integral of X to the fourth over X minus one. Here, the first thing we noticed is that the numerator has larger than read. So let's go out in these long division X minus one going into X to the fourth power we have executed. We'Ll supply that subtract left over with X cubed so we can add X squared up here we had X cubed minus X squared That's over with X squared so we could come back up here at an ex and then we'll have X squared minus x subtract left over with X And then we could come back here once more and then add that one in there and then we're X minus one Subtract that off and we had a one left over X Does not go into one. So this is our remainder. So we could come and write this as well. We just did the division our quotient was X cubed Ex player accident one and then we have our remainder divided by the question the original divisor X minus one and this is a much easier in the world. In the original, we can use the power rule for the first four terms for the last one, and may or may not help you to use the use up here. If you need to use because of the minus one, go ahead and do the U equals X minus one. So using the power rule and then the natural log, we have extra the fourth over. Four. Execute over three X squared over two X and then plus natural law. Don't forget that. Absolute value X minus one inside and closer constancy of integration and there's our answer.

To integrate this. We begin by applying integration by parts and here we want to let you be the turkana metric expression sign of forex and our D. V. Will be erased. X. Dx. So from here we get differential review equal to four times go sign of four x. DX. And our V will be equal to erased X. So by integration by parts we have this integral equal to U. Times V minus the integral of the times do you? Which is just the same as Sign of four x times erase Effexor That's erase x times sine of four x minus the integral of erased X times four times go sign of four x. And then dx Now simplifying this, we get Erased x times sine of four x -4 times the integral of erase X times go side of four X. Yes. Now from here we will apply integration by parts again. And you want to let you be the Trig and metric expression go sign of four x. And our D. V will be erased X. Dx. So from here we get differential. A view equal to negative four times sine of for X. Dx. And our V. This is equal to erased X. So from here we have Erased x times sine of four x minus four times U times V minus the integral of E. Times to you. That's just erased extent. Sign of four X minus four times U. Times V. L. B erased. X times go sign of four X minus. The integral of E. Which is erased. X. Time's D. U. Which is negative for sign of four x. and then DX. Now from here we have He raised x times sine of four x -4 times erase x times go sign of four x -16 times the integral of erase X times sine of four X. D X. Know that this is equal to the integral of erase X Times Sine of four X. Dx. So since this integral is similar to this integral, then we can combine them by adding both sides by 16 times the integral of erase two X. Time. Sign up for X. D. X. And so from there we get 17 times the integral of erased x. Times Sine of four x. DX. This is equal to erased X times sine of forex minus four times erase exco center for ex. And then pussy. and then dividing both sides by 17. We have integral of erase X. Times sine of four X. Dx. This is equal to 1/17 times the race X times sine of four X -4/17 times erase X times go center four X. And then plus C.

To integrate this, we will apply integration by parts and here you want to let you be the expression that cannot be integrated and so that will be tangent inverse of x. And devi will be the remaining factors inside this instagram that will be x squared dx. Now to find the differential a view, we simply take the differential of both sides of this equation and that will be do you equal to 1/1 plus x squared dx. And to get V we simply take the integral of both sides of this and we get the equal to x rays to the third power over three. And so by integration by parts, this is equal to U times v minus the integral of E D U. That will be tangent inverse of X times V which is x cube over three minus the integral of x cube over three times D U which is one over one plus x squared dx. Now simplifying this, we have x cubed times tangent inverse of x over three minus 1/3 integral of X cube over one plus x squared dx. Now for this integral we will apply substitution and here we want to let W equal to one plus x squared. And so the W is just equal to two X dx. Or that means one half dws X dx. Now, since the numerator is x rays to the third power and we already have X dx in terms of W. Then we need to find an expression for X squared in terms of W by manipulating this questions. So we get x squared equal to W -1. And so from here we have x rays through the third power times tangent inverse of X over three -1/3 times the integral of We have w minus one Times 1/2 dw this all over W which is equal to x cubed times tangent inverse of x over three -1/6 times 1 -1 over WDW. And so integrating this term by term, we have x cubed times tangent inverse of x Over 3 -1/6 times W minus L and absolute value of W. And then plus C. And because W is one plus X squared, then we have X cubed times tangent inverse of X over three minus 1/6 times one plus x squared minus Ln absolute value of one plus x squared or without the absolute value since one plus x ray is always positive and then plus C and Distributing the 1/6. We have X cubed times tangent inverse of X over three -1/6 minus X squared over six plus 1/6 Ln of one plus x squared and then plus C. And because 1/6 is a constant, then it will be absorbed by sea. And the final form will be x rays through the third power times, tangent inverse of X over three minus X squared over six plus 1/6 Ellen of one plus X squared, and then plus C.

Okay. Today I'm going to show you how to do the triple integral for the core sign X square in certain limited in the particular limitation. So for this one win So here is the function we need to integrate. And this is the limitation. So the tricky part is the cause An X square. So we don't have another X in this one for use three. Yeah. So we need to create another X to make the interval possible for right now. So what we can do? Let me go back to here is to why and so X is from two y two to write. So if we try to integrate y first so we can invert do you think and make the wife to be X So we left something with X So we get something we created X So let me go back to the limitation for here. So this one is why equal to half X So is here. So when we take a look so we also can He's saying he this one is x equal to two. Why x so From X equal to y to x equal to to so the area is here. So we can We can use another limitation from for act for Why? So why is from zero to why equal to two x x over two. So this one, the interview will become to this. And then if we take the d Y first we can tease. So we get actual acts here so we can use the use up tricky use of trick. So you equal to X square d'you equal to two x d x So the X equal to do you d x equal to d u uh, do I buy two X? So So for this one, we can cancel the X here, and the whole integral will become four, costing you something with some constant. Yeah, we just called this constant here and here. Also dizzy here is the constant We can calculate their and EU, so this one is easy to calculate. So you go to some constant times sign you and you just need to figure out an imitation. So for how integral we get this, I think the rest part you guys no. How do you do that? It's really formally just parking the creation and you can get the answer. Thank you.


Similar Solved Questions

5 answers
Pdf i8 f(clo) Suppose that X, Xz are iid with the Rayleigh distribution; that is, the common 20-1 Te-=2 '/0 I (x: 0), where 0 > 0 is the unknown parametcr _ Find the MLE for Is it an completeness, ancillary: Discuss the properties of this MLE; like (minimal) suflficiency, UMVUE? Why?
pdf i8 f(clo) Suppose that X, Xz are iid with the Rayleigh distribution; that is, the common 20-1 Te-=2 '/0 I (x: 0), where 0 > 0 is the unknown parametcr _ Find the MLE for Is it an completeness, ancillary: Discuss the properties of this MLE; like (minimal) suflficiency, UMVUE? Why?...
5 answers
4 (50 pts) Integrate:dx (a) xVin x e-I/r (6) / dx x2 dx (c) 4x2 dx (d) ] z+61+710 dx (e) TV16x2
4 (50 pts) Integrate: dx (a) xVin x e-I/r (6) / dx x2 dx (c) 4x2 dx (d) ] z+61+710 dx (e) TV16x2...
5 answers
The Clausius-Clapeyron equation relates vapor pressure and temperature according tot he following equation. In P = A +C RT If the vapor pressure is measured at several different temperatures,a plot of ln P us 4 can used to determine the enthalpy of vaporization, AHcap" Which of the following correctly calculate 4H rap from such a plotAHuap slope AHuap intercept on y axis AHvap slope X RstopaAHvap
The Clausius-Clapeyron equation relates vapor pressure and temperature according tot he following equation. In P = A +C RT If the vapor pressure is measured at several different temperatures,a plot of ln P us 4 can used to determine the enthalpy of vaporization, AHcap" Which of the following co...
2 answers
AxoeLEA 5 Fcna tle Vclva tue fst LLLA ocaml Scld sld Czlnuer Ad Wlck < bat 7 = 12 nded Lz t -37 and Ll plaue Xty =2
AxoeLEA 5 Fcna tle Vclva tue fst LLLA ocaml Scld sld Czlnuer Ad Wlck < bat 7 = 12 nded Lz t -37 and Ll plaue Xty =2...
5 answers
Multiply the following polynomials and simplify Your ansuer into the form anr"+n = M + do: 1-I+] and > +I +] in Rl]: ir' 20+1 + and (i-42+0 - J = Clc]: +3"+I_ Ad 2r" #+r+Lin Fskr]: Do the sume but over Fslc]:
Multiply the following polynomials and simplify Your ansuer into the form anr"+n = M + do: 1-I+] and > +I +] in Rl]: ir' 20+1 + and (i-42+0 - J = Clc]: +3"+I_ Ad 2r" #+r+Lin Fskr]: Do the sume but over Fslc]:...
5 answers
Quesdon 18Find the absolute maximum value of the function f(c) ~2 4x 2 on the interval [-3,4]:a) 00b) 0 2c) 01d) 06e) 05
Quesdon 18 Find the absolute maximum value of the function f(c) ~2 4x 2 on the interval [-3,4]: a) 00 b) 0 2 c) 01 d) 06 e) 05...
5 answers
Use cylindrical coordinates Find the volume of the solid that lies within both the cylinder x2 + y2 = 25 and the sphere x2 + y2 22 = 100.
Use cylindrical coordinates Find the volume of the solid that lies within both the cylinder x2 + y2 = 25 and the sphere x2 + y2 22 = 100....
5 answers
Question 2 (5 points) and are functions of t: Evaluate dy/dt for the following: Assume X Axy-Sx+y3_-3; dxldt =7,X-3, Y-3 Round your answer to two decimal places.Your Answer:
Question 2 (5 points) and are functions of t: Evaluate dy/dt for the following: Assume X Axy-Sx+y3_-3; dxldt =7,X-3, Y-3 Round your answer to two decimal places. Your Answer:...
5 answers
Time lel ro ualCalculate the molar_heatof fusion (4 Hmn fus) (Jmol) for the following transition Pbls) (308.45 K, atm) 'Pb(l) (807 K atm) The melting point of Pb(s) is 784.27 K and the standard molar heat of fusion is 3320 Jmol If the moler heat capacity of Pb(s) and Pb(I) are 28.2 and 45. JmolK, respectively:35549.2 b. 8887.3 17774.6 d.5924.9
Time lel ro ual Calculate the molar_heatof fusion (4 Hmn fus) (Jmol) for the following transition Pbls) (308.45 K, atm) 'Pb(l) (807 K atm) The melting point of Pb(s) is 784.27 K and the standard molar heat of fusion is 3320 Jmol If the moler heat capacity of Pb(s) and Pb(I) are 28.2 and 45. J...
5 answers
10. Which of the functions corresponds to the graph?A. f(x)=e*+2 B__ f (x)-e'+2 C. f (x)=-e*+2 D. f()=e*+1
10. Which of the functions corresponds to the graph? A. f(x)=e*+2 B__ f (x)-e'+2 C. f (x)=-e*+2 D. f()=e*+1...
5 answers
The population (In thousands) of city from 980 through 2005 can be modeled by1572e0.02/ , wherecorresponds to 1980_(a) According to this model, what was the population of the city In 2004? (Enter your answer to the nearest person:) people(b) According to this model, In what vear will the city have population of 3,400,000?
The population (In thousands) of city from 980 through 2005 can be modeled by 1572e0.02/ , where corresponds to 1980_ (a) According to this model, what was the population of the city In 2004? (Enter your answer to the nearest person:) people (b) According to this model, In what vear will the city ha...
5 answers
Quation%ipoints)Two masses and mo scparated by distance What is the force bctween two masses? How would the above force between two masses be affected if the scparation distance between them isthrice?(b) decreased by one-third?
Quation%i points) Two masses and mo scparated by distance What is the force bctween two masses? How would the above force between two masses be affected if the scparation distance between them is thrice? (b) decreased by one-third?...
5 answers
Exercise 2_ Let G be multiplicative group of order n We will Htation For example_ adopt the usual exponcntial Thcn the group G is calledl cyclic a € G such that if there exists An element {a,0.a' , (0) Show that the group Uo is a cyclic group. (b) Show that the group U,s is not cyclic group.
Exercise 2_ Let G be multiplicative group of order n We will Htation For example_ adopt the usual exponcntial Thcn the group G is calledl cyclic a € G such that if there exists An element {a,0.a' , (0) Show that the group Uo is a cyclic group. (b) Show that the group U,s is not cyclic gr...
5 answers
Amino 2-butecai2- i sopCopoxyetnetiv |nxo 3-cycopfopyicyclopentcaoi2, 3 exoxJ $ -nhrocyclonexeone 16 7O7262 Oneri 72 87v244 RMks[64 #F44e.
amino 2-butecai 2- i sopCopoxyetnetiv | nxo 3-cycopfopyicyclopentcaoi 2, 3 exoxJ $ -nhrocyclonexeone 16 7O 7262 Oneri 72 87v244 RMks[64 #F44e....

-- 0.020254--