Integrating 1/(x-11x^2) Using Partial Fraction Decomposition

The process of integrating a rational function such as 1/(x-11x^2) can be simplified by expressing the integrand in terms of simpler fractions through partial fraction decomposition. This method is particularly useful for integrating complex rational functions.

Partial Fraction Decomposition

Begin by expressing the integrand as a sum of simpler fractions:

1/(x-11x^2) A/(x-1) (Bx C)/(1x^2)

Where A, B, and C are constants to be determined.

Step 1: Setting Up the Equation

Multiplying both sides by the denominator x-11x^2:

1 A1x^2 Bx - Cx - C

Which can be rearranged as:

1 A1x^2 Bx^2 - Bx - Cx C - C

This simplifies to:

1 A Bx^2 - Bx - Cx C - C

Step 2: Equating Coefficients

Now, equate the coefficients from both sides of the equation. Since the left side has no x^2 or x terms and the constant term is 1, we have:

A B 0

-B - C 0

C - C 1

Step 3: Solving the System of Equations

From A B 0, express B in terms of A:

B -A

From -B - C 0, express C in terms of B:

C B -A

Substituting C -A into the third equation C - C 1:

A - -A 1 implies 2A 1, hence A 1/2

Using A 1/2 to find B and C:

B -A -1/2 and C -A -1/2

Step 4: Writing the Partial Fraction Decomposition

The partial fraction decomposition can now be written as:

1/(x-11x^2) 1/2(x-1) (-1/2x - 1/2)/(1x^2)

Step 5: Integrating Each Term

Now integrate each term separately:

Integrating 1/2(x-1)

This results in:

1/2 ln |x-1| C_1

Integrating -1/2x - 1/2/(1x^2)

This can be split into two integrals:

-1/2 (int x/(1x^2) dx - int 1/(1x^2) dx)

To solve the first integral, use the substitution u 1 - x^2:

int x/(1x^2) dx 1/2 ln |1 - x^2| C_2

The second integral transforms to:

int 1/(1x^2) dx -1/x C_3

Combining these results, we get:

-1/2 (1/2 ln |1 - x^2| - 1/x) C_4 -1/4 ln |1 - x^2| 1/2x C_4

Final Step: Combining the Results

Combining all the integrals, we have:

int 1/(x-11x^2) dx 1/2 ln |x-1| - 1/4 ln |1 - x^2| - 1/2x C

Final Answer: