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Find the indefinite integral using integration by partial fractions: S2x/((x^2)-4...

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kwin13 | Student, Undergraduate | (Level 1) Honors

Posted November 8, 2012 at 9:25 PM via web

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Find the indefinite integral using integration by partial fractions: S2x/((x^2)-4 dx

This is what I have figured:

2x/[(x-2)(x+2)] = A/(x-2)  + B/(x+2)

2x=A(x+2) + B(x-2)  let x=-2

2(-2) = A(-2+2) +B(-2-2)

-4=-4B  So B=1  then let x=2

2(2) = A(2+2) + B(2-2)

4=4A  So A=1

so S(1/(x-2) +1/(x+2))dx

S(1/x-2)dx + S(1/x+2)dx

ln|x-2|+ln|x+2|+C

please let me know if I am on the right track with this one.

Thanks

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted November 9, 2012 at 2:15 AM (Answer #1)

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`(2x)/(x^2-4) = (2x)/((x-2)(x+2))`

Using partial fractions;

`(2x)/((x-2)(x+2)) = A/(x-2)+B(x+2)`

            `2x = A(x+2)+B(x-2)`

            `2x = x(A+B)+2(A-B)`

 

When you consider x; in the left side the component of x (the number with x term) is 2 and the right side it is (A+B).

So;

`2 = A+B -----(1)`

 

When you consider constant term ; in the left side the component of constant is 0 and the right side it is 2(A-B).

 

`0 = 2(A-B) `

`0 = A-B ----(2)`

 

 

Once you solve (1) and (2);

`A = 1`

`B = 1`

 

So we can write;

`(2x)/((x-2)(x+2)) = 1/(x-2)+1(x+2)`

 

`int (2x)/((x-2)(x+2))dx`

`= int1/(x-2)+1(x+2) dx`

`= int 1/(x-2)dx+int1/(x+2)dx`

`= ln(x-2)+ln(x+2)+C` C is a constant.

`= ln((x-2)(x+2))+C`

 

So

`int (2x)/((x-2)(x+2))dx = ln((x-2)(x+2))+C`

 

Note:

Your method is correct. This is another way that may be easy to use with complex situations.

Sources:

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