Partial fractions

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

If the numerator degree is at least the denominator degree, divide the polynomials first. Factor the denominator and propose a sum of fractions with unknown coefficients. For repeated factors include all powers; for irreducible quadratic factors use a linear numerator. Determine coefficients and integrate term by term.

Classification reference: OpenStax · Partial fractions

Worked example

1. Identify the integrand, variables and order. For multiple integrals, start with the innermost integral.

   $$\int \frac{1}{x^{2} - 1}\, dx$$

2. Integrate with respect to x. Treat the other variables as constants at this stage.

   $$\int \frac{1}{x^{2} - 1}\, dx$$

3. Factor the denominator to identify the blocks of the decomposition.

   $$x^{2} - 1 = \left(x - 1\right) \left(x + 1\right)$$

4. Introduce unknown coefficients for each block.

   $$\frac{1}{x^{2} - 1} = \frac{A_{1}}{2 x - 2} + \frac{A_{2}}{2 x + 2}$$

5. Multiply by the common denominator and equate the coefficients of each power.

   $$\frac{A_{1}}{2} + \frac{A_{2}}{2} = 0,\quad \frac{A_{1}}{2} - \frac{A_{2}}{2} - 1 = 0$$

6. Solve the coefficient system and substitute these values into the fractions.

   $$A_{1} = 1,\quad A_{2} = -1$$

7. Decompose the rational function into partial fractions; perform polynomial division first if needed.

   $$\frac{1}{x^{2} - 1} = - \frac{1}{2 \left(x + 1\right)} + \frac{1}{2 \left(x - 1\right)}$$

8. Split the sum. The integral of a sum is the sum of its integrals.

   $$\int \left(- \frac{1}{2 \left(x + 1\right)} + \frac{1}{2 \left(x - 1\right)}\right)\, dx = \int \frac{1}{2 \left(x - 1\right)}\, dx + \int \left(- \frac{1}{2 \left(x + 1\right)}\right)\, dx$$

9. Move the constant factor outside: it does not depend on the integration variable.

   $$\int \left(- \frac{1}{2 \left(x + 1\right)}\right)\, dx = - \frac{\log{\left(x + 1 \right)}}{2}$$

10. Make a substitution and replace the differential too; both must change together.

    $$u = x + 1,\quad du = 1\,dx$$

11. For 1/x use a logarithm. On the real domain, write the primitive with an absolute value and x ≠ 0.

    $$\int \frac{1}{u}\, du = \log{\left(u \right)}$$

12. After integrating the expression in u, return to the original variable.

    $$\int \frac{1}{x + 1}\, dx = \log{\left(x + 1 \right)}$$

13. Combine the results of the substeps and simplify.

    $$\int \left(- \frac{1}{2 \left(x + 1\right)}\right)\, dx = - \frac{\log{\left(x + 1 \right)}}{2}$$

14. Move the constant factor outside: it does not depend on the integration variable.

    $$\int \frac{1}{2 \left(x - 1\right)}\, dx = \frac{\log{\left(x - 1 \right)}}{2}$$

15. Make a substitution and replace the differential too; both must change together.

    $$u = x - 1,\quad du = 1\,dx$$

16. For 1/x use a logarithm. On the real domain, write the primitive with an absolute value and x ≠ 0.

    $$\int \frac{1}{u}\, du = \log{\left(u \right)}$$

17. After integrating the expression in u, return to the original variable.

    $$\int \frac{1}{x - 1}\, dx = \log{\left(x - 1 \right)}$$

18. Combine the results of the substeps and simplify.

    $$\int \frac{1}{2 \left(x - 1\right)}\, dx = \frac{\log{\left(x - 1 \right)}}{2}$$

19. Combine the results of the substeps and simplify.

    $$\int \left(- \frac{1}{2 \left(x + 1\right)} + \frac{1}{2 \left(x - 1\right)}\right)\, dx = \frac{\log{\left(x - 1 \right)}}{2} - \frac{\log{\left(x + 1 \right)}}{2}$$

20. Check this primitive by differentiating it: recover exactly the integrand of this stage.

    $$- \frac{1}{2 \left(x + 1\right)} + \frac{1}{2 \left(x - 1\right)} = \frac{1}{x^{2} - 1}$$

21. Add the constant C: all primitives on an interval differ by a constant.

    $$\frac{\log{\left(\left|{x - 1}\right| \right)}}{2} - \frac{\log{\left(\left|{x + 1}\right| \right)}}{2} + C$$