Calculus of a Single Variable, Chapter 8, 8.5, Section 8.5, Problem 15

For the given integral problem: int (x^3+3x-4)/(x^3-4x^2+4x)dx , we may simplify by applying long division since the highest degree of x is the same from numerator and denominator side.
(x^3+3x-4)/(x^3-4x^2+4x) = 1+(4x^2-x-4)/(x^3-4x^2+4x) .
Apply partial fraction decomposition on the expression (4x^2-x-4)/(x^3-4x^2+4x) .
The pattern on setting up partial fractions will depend on the factors of the denominator. For the given problem, the factored form of the denominator will be:
(x^3-4x^2+4x) =(x)(x^2-4x+4)
=(x) (x-2)(x-2) or x(x-2)^2
For the linear factor (x) , we will have partial fraction: A/x
For the repeated linear factor (x-2)^2 , we will have partial fractions: B/(x-2) + C/(x-2)^2 .
The rational expression becomes:
(4x^2-x-4)/(x^3-4x^2+4x) =A/x +B/(x-2) + C/(x-2)^2
Multiply both side by the LCD =x(x-2)^2 :
((4x^2-x-4)/(x^3-4x^2+4x)) (x(x-2)^2)=(A/x +B/(x-2) + C/(x-2)^2)(x(x-2)^2)
4x^2-x-4=A*(x-2)^2+B*(x(x-2)) + C*x
We apply zero-factor property on x(x-2)^2 to solve for value we can assign on x.
x=0
x-2 = 0 then x=2 .
To solve for A , we plug-in x=0 :
4*0^2-0-4=A*(0-2)^2+B*(0(0-2)) + C*0
0-0-4 = A*(-2)^2 +0 +0
-4 =4A
-4/4 =(4A)/4
A =-1
To solve for C , we plug-in x=2 :
4*2^2-2-4=A*(2-2)^2+B*(2(2-2)) + C*2
16-2-4 = A*0 +B*0 +2C
10= 0 + 0 +2C
10 =2C
(10)/2= (2C)/2
C=5
To solve for B, plug-in x=1 ,A=-1 , and C=5 :

4*1^2-1-4=(-1)*(1-2)^2+B*(1(1-2)) + 5*1
4-1-4= (-1)*(-1)^2+B(1*(-1)) +5
-1= -1-B +5
-1= -B+4
-1-4= -B
-5=-B
(-5)/(-1) = (-B)/(-1)
B =5
Plug-in A = -1 , B =5, and C=5 , we get the partial decomposition:
(4x^2-x-4)/(x^3-4x^2+4x) =-1/x +5/(x-2) + 5/(x-2)^2
Then the integrand becomes:
(x^3+3x-4)/(x^3-4x^2+4x) = 1+(4x^2-x-4)/(x^3-4x^2+4x) .
=1-1/x +5/(x-2) + 5/(x-2)^2
Apply the basic integration property:int (u+-v) dx = int (u) dx +- int (v) dx .
int (x^3+3x-4)/(x^3-4x^2+4x) dx = int [1-1/x +5/(x-2) + 5/(x-2)^2] dx
=int1 dx - int 1/x dx +int 5/(x-2)dx + int 5/(x-2)^2dx

Apply basic integration property: int(a) dx = ax+C
int1 dx = 1x or x
Apply integration formula for logarithm: int 1/u du = ln|u|+C .
int 1/x dx=ln|x|
int 5/(x-2)dx= int 5/udu
= 5ln|u|
=5 ln|x-2|
Note: Let u =x-2 then du = dx .
Apply the Power Rule for integration: int (u^n) dx =u^(n+1)/ (n+1) +C .
int 5/(x-2)^2dx=int 5/u^2du
=int 5u^(-2)du
= 5 * u^(-2+1)/(-2+1)
= 5* u^-1/(-1)
= -5/u
= -5/(x-2)
Note: Let u =x-2 then du = dx
Combining the results, we get the indefinite integral as:
int (x^3+3x-4)/(x^3-4x^2+4x)dx =x-ln|x| +5 ln|x-2|-5/(x-2)+C

Comments

Post a Comment

Popular posts from this blog

How does Bilbo show leadership and courage in The Hobbit?

Bilbo’s increase in confidence and courage is the defining aspect of his character arc. At the beginning of the novel, he is shut away content in his Hobbit hole, happy to remain separate and coddled from the rest of the world. In the movie, Bilbo chooses to join the dwarves on their quest out of a sense for adventure. In the book, however, Bilbo wants nothing to do with them and only joins because he had been duped into signing a contract. Despite being on the adventure somewhat against his will, Bilbo still proves himself time and again, saving the dwarves from the spiders, the trolls, and a number of other dangers they encounter along the way. Bilbo’s character arc has even been cited as a sort of Christian metaphor for the growth of the soul, though how much Tolkien intended on this front is up for debate. From a thematic standpoint, the ring of invisibility also plays into Bilbo’s brand of bravery. If Hobbits are the overlooked creatures of Middle Earth, then it is noteworthy that...
Read more

In “Goodbye to All That,” Joan Didion writes that the “lesson” of her story is that “it is distinctly possible to remain too long at the fair.” What does she mean? How does the final section of the essay portray how she came to this understanding, her feelings about it, and the consequences of it?

In this classic essay, Didion describes moving to New York City from her native Sacramento at the age of twenty-one, and, as she freely admits, a somewhat naive twenty-one, her mind, "programmed by all the movies I had ever seen and all the songs I had ever heard about New York," with the typical feeling, at that age, that, "nothing like this...has ever happened to anyone before." Intending to stay for only six months, she falls in love with the city. As she says, "I do not mean 'love' in any colloquial way, I mean that I was in love with the city the way you love the first person who ever touches you and never love anyone quite that way again." This initial love led her to prolong her residency, since, "I still believed in possibilities then, still had the sense, so peculiar to New York, that something extraordinary would happen any minute, any day, any month." But as the next years roll past, and the "new faces" she encounters...
Read more

Why does the poet say "all the men and women merely players"?

It's one of the most famous lines in literature: All the world's a stage,And all the men and women merely players . . . This famous saying comes from Shakespeare's play As You Like It, and it's spoken by Jacques. What he's saying here is that simply going about our daily lives is a type of performance. Just like actors on a stage, we all have roles to perform: Daughter. Student. Sister. Captain of the soccer team. The fact that Shakespeare writes that men and women are merely players suggests that the roles are preordained and that a person's choice in being a mother, for example, or a doctor or starting point guard for the Brooklyn Nets is out of their hands. What roles a person performs over the course of their lifetime is up to fate. The notion that "all the world's a stage" has entered pop culture, but within the context of the play, this idea does not sit well with Jacques. He's a pessimist, and the idea that everything is up to fate makes...
Read more