A Question of integration based on partial fraction but my question is why does it work?

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Integrate
$$
frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)
$$
Now in this problem we divide it into
$$
fracAx-4+fracBx-5+fracCx-6+1
$$
My question is: even if we do not consider $+1$ the values of A B and C comes the same so what does that mean? Why does this happen?

integration

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edited Jul 15 at 6:52

enzotib

5,72321430

asked Jul 15 at 6:30

bhaveshgoel07

1

add a comment | 

up vote
-2
down vote

favorite

Integrate
$$
frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)
$$
Now in this problem we divide it into
$$
fracAx-4+fracBx-5+fracCx-6+1
$$
My question is: even if we do not consider $+1$ the values of A B and C comes the same so what does that mean? Why does this happen?

integration

share|cite|improve this question

edited Jul 15 at 6:52

enzotib

5,72321430

asked Jul 15 at 6:30

bhaveshgoel07

1

add a comment | 

up vote
-2
down vote

favorite

up vote
-2
down vote

favorite

Integrate
$$
frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)
$$
Now in this problem we divide it into
$$
fracAx-4+fracBx-5+fracCx-6+1
$$
My question is: even if we do not consider $+1$ the values of A B and C comes the same so what does that mean? Why does this happen?

integration

share|cite|improve this question

edited Jul 15 at 6:52

enzotib

5,72321430

asked Jul 15 at 6:30

bhaveshgoel07

1

Integrate
$$
frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)
$$
Now in this problem we divide it into
$$
fracAx-4+fracBx-5+fracCx-6+1
$$
My question is: even if we do not consider $+1$ the values of A B and C comes the same so what does that mean? Why does this happen?

integration

share|cite|improve this question

edited Jul 15 at 6:52

enzotib

5,72321430

asked Jul 15 at 6:30

bhaveshgoel07

1

share|cite|improve this question

share|cite|improve this question

edited Jul 15 at 6:52

enzotib

5,72321430

edited Jul 15 at 6:52

enzotib

5,72321430

edited Jul 15 at 6:52

enzotib

5,72321430

5,72321430

asked Jul 15 at 6:30

bhaveshgoel07

1

asked Jul 15 at 6:30

bhaveshgoel07

1

asked Jul 15 at 6:30

bhaveshgoel07

1

1

add a comment | 

add a comment | 

2 Answers
2

active

oldest

votes

up vote
1
down vote

If you have
$$fracAx-4+fracBx-5+fracCx-6tag*$$
it will simplify to something of the form
$$fracRx^2+Sx+T(x-4)(x-5)(x-6)$$
with a quadratic in the numerator. But your
integrand has a cubic in the numerator, so cannot be written
in the form (*).

share|cite|improve this answer

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 ensures coefficient of x in the denominator matches with that of the numerator.
  – Eval
  Jul 15 at 6:46
  

  
  
  
  

add a comment | 

up vote
0
down vote

$$frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=A/(x-4)+B/(x-5)+C/(x-6)+1$$

is the correct form. Note that the degree of both top and bottom is the same so we first divide to get a $1$ as the quotient.

If you do not include a $+1$ on the RHS then as $x$ approached infinity, the right side approaches $0$ while the left side approaches $1$

Note that with the wrong form you may still find A, B, C but the answer is wrong.

share|cite|improve this answer

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

add a comment | 

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

If you have
$$fracAx-4+fracBx-5+fracCx-6tag*$$
it will simplify to something of the form
$$fracRx^2+Sx+T(x-4)(x-5)(x-6)$$
with a quadratic in the numerator. But your
integrand has a cubic in the numerator, so cannot be written
in the form (*).

share|cite|improve this answer

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 ensures coefficient of x in the denominator matches with that of the numerator.
  – Eval
  Jul 15 at 6:46
  

  
  
  
  

add a comment | 

up vote
1
down vote

If you have
$$fracAx-4+fracBx-5+fracCx-6tag*$$
it will simplify to something of the form
$$fracRx^2+Sx+T(x-4)(x-5)(x-6)$$
with a quadratic in the numerator. But your
integrand has a cubic in the numerator, so cannot be written
in the form (*).

share|cite|improve this answer

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 ensures coefficient of x in the denominator matches with that of the numerator.
  – Eval
  Jul 15 at 6:46
  

  
  
  
  

add a comment | 

up vote
1
down vote

up vote
1
down vote

If you have
$$fracAx-4+fracBx-5+fracCx-6tag*$$
it will simplify to something of the form
$$fracRx^2+Sx+T(x-4)(x-5)(x-6)$$
with a quadratic in the numerator. But your
integrand has a cubic in the numerator, so cannot be written
in the form (*).

share|cite|improve this answer

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

If you have
$$fracAx-4+fracBx-5+fracCx-6tag*$$
it will simplify to something of the form
$$fracRx^2+Sx+T(x-4)(x-5)(x-6)$$
with a quadratic in the numerator. But your
integrand has a cubic in the numerator, so cannot be written
in the form (*).

share|cite|improve this answer

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

share|cite|improve this answer

share|cite|improve this answer

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

answered Jul 15 at 6:34

Lord Shark the Unknown

85.8k951112

85.8k951112

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 ensures coefficient of x in the denominator matches with that of the numerator.
  – Eval
  Jul 15 at 6:46
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  +1 ensures coefficient of x in the denominator matches with that of the numerator.
  – Eval
  Jul 15 at 6:46
  

  
  
  
  

+1 ensures coefficient of x in the denominator matches with that of the numerator.
– Eval
Jul 15 at 6:46

+1 ensures coefficient of x in the denominator matches with that of the numerator.
– Eval
Jul 15 at 6:46

add a comment | 

up vote
0
down vote

$$frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=A/(x-4)+B/(x-5)+C/(x-6)+1$$

is the correct form. Note that the degree of both top and bottom is the same so we first divide to get a $1$ as the quotient.

If you do not include a $+1$ on the RHS then as $x$ approached infinity, the right side approaches $0$ while the left side approaches $1$

Note that with the wrong form you may still find A, B, C but the answer is wrong.

share|cite|improve this answer

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

add a comment | 

up vote
0
down vote

$$frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=A/(x-4)+B/(x-5)+C/(x-6)+1$$

is the correct form. Note that the degree of both top and bottom is the same so we first divide to get a $1$ as the quotient.

If you do not include a $+1$ on the RHS then as $x$ approached infinity, the right side approaches $0$ while the left side approaches $1$

Note that with the wrong form you may still find A, B, C but the answer is wrong.

share|cite|improve this answer

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

add a comment | 

up vote
0
down vote

up vote
0
down vote

$$frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=A/(x-4)+B/(x-5)+C/(x-6)+1$$

is the correct form. Note that the degree of both top and bottom is the same so we first divide to get a $1$ as the quotient.

If you do not include a $+1$ on the RHS then as $x$ approached infinity, the right side approaches $0$ while the left side approaches $1$

Note that with the wrong form you may still find A, B, C but the answer is wrong.

share|cite|improve this answer

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

$$frac(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=A/(x-4)+B/(x-5)+C/(x-6)+1$$

is the correct form. Note that the degree of both top and bottom is the same so we first divide to get a $1$ as the quotient.

If you do not include a $+1$ on the RHS then as $x$ approached infinity, the right side approaches $0$ while the left side approaches $1$

Note that with the wrong form you may still find A, B, C but the answer is wrong.

share|cite|improve this answer

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

share|cite|improve this answer

share|cite|improve this answer

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

answered Jul 15 at 6:45

Mohammad Riazi-Kermani

27.6k41852

27.6k41852

add a comment | 

add a comment | 

 

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