Integration of rational functions-Integration Formulas

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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 a d x = a x + C

2. Integral of x

\int x d x = \frac{x^{2}}{2} + C

3. Integral of x²

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

4. Integral of the power function

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

5. Integral of a linear function raised to nth power

\int (a x + b)^{n} d x = \frac{(a x + b)^{n + 1}}{a (n + 1)} + C, n \neq - 1

6. Integral of the reciprocal function

\int \frac{d x}{x} = \ln | x | + C

7. Integral of a rational function with a linear denominator

\int \frac{d x}{a x + b} = \frac{1}{a} \ln | a x + b | + C

8. Integral of a linear fractional function

\int \frac{a x + b}{c x + d} = \frac{a}{c} x + \frac{b c - a d}{c^{2}} \ln | c x + d | + C

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

10 . \int \frac{d x}{a + b x} = \frac{1}{b^{2}} (a + b x - a \ln | a + b x |) + C

11 . \int \frac{x^{2} d x}{a + b x} = \frac{1}{b^{3}} [\frac{1}{2} (a + b x)^{2} - 2 a (a + b x) + a^{2} \ln | a + b x |] + 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{x d x}{(a + b x)^{2}} = \frac{1}{b^{2}} (\ln | a + b x | + \frac{a}{a + b x}) + C

15 . \int \frac{x^{2} d x}{(a + b x)^{2}} = \frac{1}{b^{2}} (a + b x - 2 a \ln | a + b x | - \frac{a^{2}}{a + b x}) + 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{d x}{1 + x^{2}} = \tan^{- 1} x + C

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

23 . \int \frac{x d x}{a^{2} + x^{2}} = \frac{1}{2} \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{d x}{a x^{2} + b x + c} = \frac{1}{\sqrt{b^{2} - 4 a c}} \ln \left | \frac{2 a x + b - \sqrt{b^{2} - 4 a c}}{2 a x + b + \sqrt{b^{2} - 4 a c}} \right | + C, D = b^{2} - 4 a c > 0

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

\int \frac{d x}{a x^{2} + b x + c} = \frac{1}{\sqrt{4 a c - b^{2}}} \tan^{- 1} \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} + C, D = b^{2} - 4 a c \lt 0