## Integration of rational functions-Integration Formulas

### 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

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

### 6. Integral of the reciprocal function

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

### 7. Integral of a rational function with a linear denominator

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

### 8. Integral of a linear fractional function

$\int \frac{ax+b}{cx+d}=\frac{a}{c}x+\frac{bc–ad}{{c}^{2}}\mathrm{ln}|cx+d|+C$
$$9. \ \int \frac {dx}{(x+a)(x+b)} = \frac {1}{a-b} \ln \left \vert \frac {x+b}{x+a} \right \vert +C, \ a \neq b$$
$$12. \ \int \frac {dx}{x(a+bx)} = \frac 1a \ln \left \vert \frac {a+bx}{x} \right \vert +C$$
$$13. \ \int \frac {dx}{x^2(a+bx)} = – \frac {1}{ax} + \frac {b}{a^2} \ln \left \vert \frac {a+bx}{x} \right \vert +C$$
$$16. \ \int \frac {dx}{x(a+bx)^2} = \frac {1}{a(a+bx)} + \frac {1}{a^2} \ln \left \vert \frac {a+bx}{x} \right \vert +C$$
$$17. \ \int \frac {dx}{x^2-1} = \frac 12 \ln \left \vert \frac {x-1}{x+1} \right \vert +C$$
$$18. \ \int \frac {dx}{1-x^2} = \frac 12 \ln \left \vert \frac {1+x}{1-x} \right \vert +C$$
$$19. \ \int \frac {dx}{a^2-x^2} = \frac {1}{2a} \ln \left \vert \frac {a+x}{a-x} \right \vert +C$$
$$20. \ \int \frac {dx}{x^2-a^2} = \frac {1}{2a} \ln \left \vert \frac {x-a}{x+a} \right \vert +C$$
$$24. \ \int \frac {dx}{a+bx^2} = \frac {1}{\sqrt ab} \tan^{-1} \left( x \sqrt \frac ba \right) +C, \ ab \gt 0$$
$$25. \ \int \frac {xdx}{a+bx^2} = \frac {1}{2b} \ln \left \vert x^2 + \frac ab \right \vert +C$$
$$26. \ \int \frac {dx}{x(a+bx^2)} = \frac {1}{2a} \ln \left \vert \frac {x^2}{a+bx^2} \right \vert +C$$
$$27. \ \int \frac {dx}{a^2+b^2x^2} = \frac {1}{2ab} \ln \left \vert \frac {a+bx}{a-bx} \right \vert +C$$