The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:
which can then be integrated term by term. For other types of functions that can be integrated, see lists of integrals.
Miscellaneous integrands
Integrands of the form xm(a x + b)n Many of the following antiderivatives have a term of the form ln |ax + b|. Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function. However, it is conventional to omit this from the notation. For example,
∫ 1 a x + b d x = { 1 a ln ( − ( a x + b ) ) + C − a x + b < 0 1 a ln ( a x + b ) + C + a x + b > 0 {\displaystyle \int {\frac {1}{ax+b}}\,dx={\begin{cases}{\dfrac {1}{a}}\ln(-(ax+b))+C^{-}&ax+b<0\\{\dfrac {1}{a}}\ln(ax+b)+C^{+}&ax+b>0\end{cases}}}
is usually abbreviated as
∫ 1 a x + b d x = 1 a ln | a x + b | + C , {\displaystyle \int {\frac {1}{ax+b}}\,dx={\frac {1}{a}}\ln \left|ax+b\right|+C,}
where C is to be understood as notation for a locally constant function of x. This convention will be adhered to in the following.
Integrands of the form xm / (a x2 + b x + c)n For a ≠ 0 : {\displaystyle a\neq 0:}
Integrands of the form xm (a + b xn)p The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents.
Integrands of the form (A + B x) (a + b x)m (c + d x)n (e + f x)p The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, n and p toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents. Special cases of these reductions formulas can be used for integrands of the form ( a + b x ) m ( c + d x ) n ( e + f x ) p {\displaystyle (a+b\,x)^{m}(c+d\,x)^{n}(e+f\,x)^{p}} by setting B to 0.
Integrands of the form xm (A + B xn) (a + b xn)p (c + d xn)q The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, p and q toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents. Special cases of these reductions formulas can be used for integrands of the form ( a + b x n ) p ( c + d x n ) q {\displaystyle \left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)^{q}} and x m ( a + b x n ) p ( c + d x n ) q {\displaystyle x^{m}\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)^{q}} by setting m and/or B to 0.
Integrands of the form (d + e x)m (a + b x + c x2)p when b2 − 4 a c = 0 The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents. Special cases of these reductions formulas can be used for integrands of the form ( a + b x + c x 2 ) p {\displaystyle \left(a+b\,x+c\,x^{2}\right)^{p}} when b 2 − 4 a c = 0 {\displaystyle b^{2}-4\,a\,c=0} by setting m to 0.
Integrands of the form (d + e x)m (A + B x) (a + b x + c x2)p The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents. Special cases of these reductions formulas can be used for integrands of the form ( a + b x + c x 2 ) p {\displaystyle \left(a+b\,x+c\,x^{2}\right)^{p}} and ( d + e x ) m ( a + b x + c x 2 ) p {\displaystyle (d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}} by setting m and/or B to 0.
Integrands of the form xm (a + b xn + c x2n)p when b2 − 4 a c = 0 The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents. Special cases of these reductions formulas can be used for integrands of the form ( a + b x n + c x 2 n ) p {\displaystyle \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}} when b 2 − 4 a c = 0 {\displaystyle b^{2}-4\,a\,c=0} by setting m to 0.
Integrands of the form xm (A + B xn) (a + b xn + c x2n)p The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents. Special cases of these reductions formulas can be used for integrands of the form ( a + b x n + c x 2 n ) p {\displaystyle \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}} and x m ( a + b x n + c x 2 n ) p {\displaystyle x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}} by setting m and/or B to 0.
References