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Calculus

# Integrals of Rational Functions

Argument (independent variable): $$x$$
Real numbers: $$C$$, $$a$$, $$b$$, $$c$$, $$p$$, $$n$$

Discriminant of a quadratic equation: $$D$$

1. 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. Below we consider a list of the most common integrals of rational functions.
2. Integral of a constant
$${\large\int\normalsize} {adx} = ax + C$$
3. Integral of $$x$$
$${\large\int\normalsize} {xdx} = {\large\frac{{{x^2}}}{2}\normalsize} + C$$
4. Integral of$${x^2}$$
$${\large\int\normalsize} {{x^2}dx} = {\large\frac{{{x^3}}}{3}\normalsize} + C$$
5. Integral of the power function
$${\large\int\normalsize} {{x^p}dx} = {\large\frac{{{x^{p + 1}}}}{{p + 1}}\normalsize} + C,\;$$ $$p \ne – 1.$$
6. Integral of a linear function raised to $$n$$th power
$${\large\int\normalsize} {{{\left( {ax + b} \right)}^n}dx} =$$ $${\large\frac{{{{\left( {ax + b} \right)}^{n + 1}}}}{{a\left( {n + 1} \right)}}\normalsize} + C,\;$$ $$n \ne – 1.$$
7. Integral of the reciprocal function
$${\large\int {\frac{{dx}}{x}}\normalsize} = \ln \left| x \right| + C$$
8. Integral of a rational function with a linear denominator
$${\large\int {\frac{{dx}}{{ax + b}}}\normalsize} =$$ $${\large\frac{1}{a}\normalsize}\ln \left| {ax + b} \right| + C$$
9. Integral of a linear fractional function
$${{{\large\int\normalsize}{\large\frac{{ax + b}}{{cx + d}}\normalsize}} \,dx} =$$ $${\large\frac{a}{c}\normalsize} x \;+$$ $${\large\frac{{bc – ad}}{{{c^2}}}\,\normalsize} \ln \left| {cx + d} \right|$$ $$+\;C$$
10. $${\large\int {\frac{{dx}}{{\left( {x + a} \right)\left( {x + b} \right)}}}\normalsize} =$$ $${\large\frac{1}{{a – b}}\normalsize} \,\ln \left| {\large\frac{{x + b}}{{x + a}}\normalsize} \right| + C,\;$$ $$a \ne b.$$
11. $${\large\int {\frac{{dx}}{{a + bx}}}\normalsize} =$$ $${\large\frac{1}{{{b^2}}}\normalsize}\big( {a + bx} \;-$$ $${ a\ln \left| {a + bx} \right|} \big) + C$$
12. $${\large\int {\frac{{{x^2}dx}}{{a + bx}}}\normalsize} =$$ $${\large\frac{1}{{{b^3}}}\normalsize}\Big[ {\large\frac{1}{2}\normalsize{{\left( {a + bx} \right)}^2} \;}-$$ $${2a\left( {a + bx} \right) }\;+$$ $${{a^2}\ln \left| {a + bx} \right|} \Big] + C$$
13. $${\large\int {\frac{{dx}}{{x\left( {a + bx} \right)}}}\normalsize} =$$ $${\large\frac{1}{a}\normalsize}\ln \left| {{\large\frac{{a + bx}}{x}}\normalsize} \right| + C$$
14. $${\large\int {\frac{{dx}}{{{x^2}\left( {a + bx} \right)}}}\normalsize} =$$ $$– {\large\frac{1}{{ax}}\normalsize} + {\large\frac{b}{{{a^2}}}\normalsize}\ln \left| {{\large\frac{{a + bx}}{x}}\normalsize} \right| + C$$
15. $${\large\int {\frac{{xdx}}{{{{\left( {a + bx} \right)}^2}}}}\normalsize} =$$ $${\large\frac{1}{{{b^2}}}\normalsize}\left( {\ln \left| {a + bx} \right| + {\large\frac{a}{{a + bx}}\normalsize}} \right)$$ $$+\; C$$
16. $${\large\int {\frac{{{x^2}dx}}{{{{\left( {a + bx} \right)}^2}}}}\normalsize} =$$ $${\large\frac{1}{{{b^3}}}\normalsize}\Big( {a + bx }\;-$$ $${2a\ln \left| {a + bx} \right| \;}-$$ $${{\large\frac{{{a^2}}}{{a + bx}}\normalsize}} \Big) + C$$
17. $${\large\int {\frac{{dx}}{{x{{\left( {a + bx} \right)}^2}}}}\normalsize} =$$ $${\large\frac{1}{{a\left( {a + bx} \right)}}\normalsize} \;+$$ $${\large\frac{1}{{{a^2}}}\normalsize}\ln \left| {{\large\frac{{a + bx}}{x}}\normalsize} \right| + C$$
18. $${\large\int {\frac{{dx}}{{{x^2} – 1}}}\normalsize} =$$ $${\large\frac{1}{2}\normalsize}\ln \left| {\large\frac{{x – 1}}{{x + 1}}\normalsize} \right| + C$$
19. $${\large\int {\frac{{dx}}{{1 – {x^2}}}}\normalsize} =$$ $${\large\frac{1}{2}\normalsize}\ln \left| {\large\frac{{1 + x}}{{1 – x}}\normalsize} \right| + C$$
20. $${\large\int {\frac{{dx}}{{{a^2} – {x^2}}}}\normalsize} =$$ $${\large\frac{1}{2a}\normalsize}\ln \left| {\large\frac{{a + x}}{{a – x}}\normalsize} \right| + C$$
21. $${\large\int {\frac{{dx}}{{{x^2} – {a^2}}}}\normalsize} =$$ $${\large\frac{1}{2a}\normalsize}\ln \left| {\large\frac{{x – a}}{{x + a}}\normalsize} \right| + C$$
22. $${\large\int {\frac{{dx}}{{1 + {x^2}}}}\normalsize} =$$ $$\arctan x + C$$
23. $${\large\int {\frac{{dx}}{{{a^2} + {x^2}}}}\normalsize} =$$ $${\large\frac{1}{a}\normalsize}\arctan {\large\frac{x}{a}\normalsize} + C$$
24. $${\large\int {\frac{{xdx}}{{{a^2} + {x^2}}}}\normalsize} =$$ $${\large\frac{1}{2}\normalsize} \ln\left( {{x^2} + {a^2}} \right) + C$$
25. $${\large\int {\frac{{dx}}{{a + b{x^2}}}}\normalsize} =$$ $${\large\frac{1}{{\sqrt {ab} }}\normalsize} \arctan\left( {x\sqrt {\large\frac{b}{a}}\normalsize } \right) + C,\;$$ $$ab \gt 0.$$
26. $${\large\int {\frac{{xdx}}{{a + b{x^2}}}}\normalsize} =$$ $${\large\frac{1}{{2b}}\normalsize} \ln\left| {{x^2} + {\large\frac{a}{b}\normalsize}} \right| + C$$
27. $${\large\int {\frac{{dx}}{{x\left( {a + b{x^2}}\right)\normalsize}}}} =$$ $${\large\frac{1}{{2a}}\normalsize} \ln\left| {\large\frac{{{x^2}}}{{a + b{x^2}}}\normalsize} \right| + C$$
28. $${\large\int {\frac{{dx}}{{{a^2} – {b^2}{x^2}}}}\normalsize} =$$ $${\large\frac{1}{{2ab}}\,\normalsize} \ln\left| {\large\frac{{a + bx}}{{a – bx}}\normalsize} \right| + C$$
29. Integral of a rational function with a quadratic denominator (the case of a positive discriminant)
$${\large\int {\frac{{dx}}{{a{x^2} + bx + c}}}\normalsize} =$$ $${\large\frac{1}{{\sqrt {{b^2} – 4ac} }}\normalsize} \ln\left| {{\large\frac{{2ax + b – \sqrt {{b^2} – 4ac} }}{{2ax + b + \sqrt {{b^2} – 4ac} }}}\normalsize} \right|$$ $$+\; C,\;$$ $$D = {b^2} – 4ac \gt 0.$$
30. Integral of a rational function with a quadratic denominator (the case of a negative discriminant)
$${\large\int {\frac{{dx}}{{a{x^2} + bx + c}}}\normalsize} =$$ $${\large\frac{1}{{\sqrt {4ac – {b^2}} }}\normalsize} \arctan{\large\frac{{2ax + b}}{{\sqrt {4ac – {b^2}} }}\normalsize}$$ $$+\;C,\;$$ $$D = {b^2} – 4ac \lt 0.$$