Did I do this properly? Integration by Partial Fractions

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Homework Help Overview

The discussion revolves around evaluating the indefinite integral of the function (6x + 7)/(x^2 + 1) using integration techniques, specifically partial fractions.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply partial fraction decomposition to the integral but faces challenges in their calculations and assumptions regarding the factorization of the denominator.
  • Some participants question the validity of using partial fractions for this integral and suggest alternative approaches, such as separating the integral into two simpler parts.
  • One participant expresses confusion regarding their own calculations and seeks clarification on their method.

Discussion Status

Participants are actively engaging with the original poster's approach, providing feedback and questioning the assumptions made about the factorization of the denominator. There is no explicit consensus on the correctness of the original poster's solution, but alternative methods are being explored.

Contextual Notes

There appears to be a misunderstanding regarding the factorization of x^2 + 1, which is not factorizable into real linear factors. This misunderstanding is central to the discussion.

the7joker7
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Homework Statement



Evaluate the indefinite integral.

int (6 x + 7)/(x^2 + 1) dx `

The Attempt at a Solution



A/(x + 1) + B/(x - 1)

6x + 7 = A(x - 1) + B(x + 1)

6x + 7 = (A + B)x + (-A + B)

A + B = 6

-A + B = 7

A + (7 + A) = 6

2A = -1.

A = -.5

B = 3.5

So the answer should be...

-.5ln(x + 1) + 3.5ln(x - 1) + C

Is this correct?
 
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Differentiate your anti-derivative and see if it becomes your original Integral.

Also, I don't think it is. How did you break ... [tex]x^2+1[/tex] ?

You don't need to use Partial Fractions.

[tex]\int\frac{6x+7}{x^2+1}dx[/tex][tex]\int\left(\frac{6x}{x^2+1}+\frac{7}{x^2+1}\right)dx[/tex]
 
Last edited:
-.5(1/x + 1) + 3.5(1/x - 1)

Blah. =/

What did I do wrong?
 
Looking at your work ... looks like you broke [tex]x^2+1[/tex] to [tex](x+1)(x-1)[/tex]

Yes?

[tex]x^2-1=(x+1)(x-1)[/tex] so ... [tex]x^2+1\neq(x+1)(x-1)[/tex]
 

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