Integration of rational functions

A function or fraction is called rational if it is represented as a ratio of two polynomials. A rational function is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.

Argument (independent variable): x

Discriminant of a quadratic equation: D

Real numbers: C, a, b, c, p, n

1. Integral of a constant

$$ \int adx=ax+C $$

2. Integral of x

$$ \int xdx= \frac {x^2}{2} + C $$

3. Integral of x²

$$ \int x^2dx= \frac {x^3}{3} + C $$

4. Integral of the power function

$$ \int x^pdx= \frac {x^{p+1}}{p+1} +C, \ p \neq -1 $$

5. Integral of a linear function raised to nth power

$$ \int (ax+b)^ndx= \frac {(ax+b)^{n+1}}{a(n+1)} +C, \ n \neq -1 $$

6. Integral of the reciprocal function

$$ \int \frac {dx}{x} = \ln \vert x \vert +C $$

7. Integral of a rational function with a linear denominator

$$ \int \frac {dx}{ax+b} = \frac 1a \ln \vert ax+b \vert +C $$

8. Integral of a linear fractional function

$$ \int \frac {ax+b}{cx+d} = \frac ac x + \frac {bc-ad}{c^2} \ln \vert cx+d \vert +C $$

$$ 9. \ \int \frac {dx}{(x+a)(x+b)} = \frac {1}{a-b} \ln \left \vert \frac {x+b}{x+a} \right \vert +C, \ a \neq b $$

$$ 10. \ \int \frac {dx}{a+bx} = \frac {1}{b^2} \left( a+bx-a \ln \vert a+bx \vert \right) +C $$

$$ 11. \ \int \frac {x^2dx}{a+bx} = \frac {1}{b^3} \left[ \frac 12 (a+bx)^2 -2a(a+bx) +a^2 \ln \vert a+bx \vert \right] +C $$

$$ 12. \ \int \frac {dx}{x(a+bx)} = \frac 1a \ln \left \vert \frac {a+bx}{x} \right \vert +C $$

$$ 13. \ \int \frac {dx}{x^2(a+bx)} = - \frac {1}{ax} + \frac {b}{a^2} \ln \left \vert \frac {a+bx}{x} \right \vert +C $$

$$ 14. \ \int \frac {xdx}{(a+bx)^2} = \frac {1}{b^2} \left( \ln \vert a+bx \vert + \frac {a}{a+bx} \right) +C $$

$$ 15. \ \int \frac {x^2dx}{(a+bx)^2} = \frac {1}{b^2} \left( a+ bx -2a \ln \vert a+bx \vert - \frac {a^2}{a+bx} \right) +C $$

$$ 16. \ \int \frac {dx}{x(a+bx)^2} = \frac {1}{a(a+bx)} + \frac {1}{a^2} \ln \left \vert \frac {a+bx}{x} \right \vert +C $$

$$ 17. \ \int \frac {dx}{x^2-1} = \frac 12 \ln \left \vert \frac {x-1}{x+1} \right \vert +C $$

$$ 18. \ \int \frac {dx}{1-x^2} = \frac 12 \ln \left \vert \frac {1+x}{1-x} \right \vert +C $$

$$ 19. \ \int \frac {dx}{a^2-x^2} = \frac {1}{2a} \ln \left \vert \frac {a+x}{a-x} \right \vert +C $$

$$ 20. \ \int \frac {dx}{x^2-a^2} = \frac {1}{2a} \ln \left \vert \frac {x-a}{x+a} \right \vert +C $$

$$ 21. \ \int \frac {dx}{1+x^2} = \tan^{-1}x +C $$

$$ 22. \ \int \frac {dx}{a^2+x^2} = \frac 1a \tan^{-1} \frac xa +C $$

$$ 23. \ \int \frac {xdx}{a^2+x^2} = \frac 12 \ln (a^2+x^2) +C $$

$$ 24. \ \int \frac {dx}{a+bx^2} = \frac {1}{\sqrt ab} \tan^{-1} \left( x \sqrt \frac ba \right) +C, \ ab \gt 0 $$

$$ 25. \ \int \frac {xdx}{a+bx^2} = \frac {1}{2b} \ln \left \vert x^2 + \frac ab \right \vert +C $$

$$ 26. \ \int \frac {dx}{x(a+bx^2)} = \frac {1}{2a} \ln \left \vert \frac {x^2}{a+bx^2} \right \vert +C $$

$$ 27. \ \int \frac {dx}{a^2+b^2x^2} = \frac {1}{2ab} \ln \left \vert \frac {a+bx}{a-bx} \right \vert +C $$

28. Integral of a rational function with a quadratic denominator (the case of a positive discriminant)

$$ \int \frac {dx}{ax^2+bx+c} = \frac {1}{ \sqrt {b^2-4ac}} \ln \left \vert \frac {2ax+b- \sqrt {b^2-4ac} }{2ax+b+ \sqrt {b^2-4ac}} \right \vert +C, \ D=b^2-4ac \gt 0 $$

29. Integral of a rational function with a quadratic denominator (the case of a negative discriminant)

$$ \int \frac {dx}{ax^2+bx+c} = \frac {1}{ \sqrt {4ac-b^2}} \tan^{-1} \frac {2ax+b}{\sqrt {4ac-b^2}} +C, \ D=b^2-4ac \lt 0 $$