From Wikipedia, the free encyclopedia
The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.
Integrals involving r = √a2 + x2
[edit]
![{\displaystyle \int r\,dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left(x+r\right)\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83NTdmZjllYzAzNTQ4YmMyNjdiYWExZTliNjJhZTA5YmEyOTk1OWNi&__proxy_form=0)
![{\displaystyle \int r^{3}\,dx={\frac {1}{4}}xr^{3}+{\frac {3}{8}}a^{2}xr+{\frac {3}{8}}a^{4}\ln \left(x+r\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83YWE2MzczOWMwMGExNzYzZTg4YzNlM2UxMWE5NTYyM2QxZGMyZDdm&__proxy_form=0)
![{\displaystyle \int r^{5}\,dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left(x+r\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iMzYxNmQ2YzM4ODk3MDdlNDQwNWIyM2U2YjM4MDMxZTU3ZDkxZWQ4&__proxy_form=0)
![{\displaystyle \int xr\,dx={\frac {r^{3}}{3}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lYmI3MTQ1MmRiYTU0MGNiYmI3ZTk0NDI2YjYzZmI1MjY4YzQ3Yzdm&__proxy_form=0)
![{\displaystyle \int xr^{3}\,dx={\frac {r^{5}}{5}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83NzAxOWRkMmFmZDgyOTIzYTk3M2FhMmY3NjgwYTk3ZTNkZjBhZWE4&__proxy_form=0)
![{\displaystyle \int xr^{2n+1}\,dx={\frac {r^{2n+3}}{2n+3}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83ZmE1NDU1NTRiMzg0ZjMzMjUyMTIzNjY5NmVmYzI3ZjhiNTBkOTc3&__proxy_form=0)
![{\displaystyle \int x^{2}r\,dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left(x+r\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81OWM5MzQ5ZDIyOWFjODA4ZDEzYTk3ZTA3ZmI1MWE5NTA4YjZlNWUz&__proxy_form=0)
![{\displaystyle \int x^{2}r^{3}\,dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left(x+r\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wYmQzODNhYjRlYjNkZWU3NTdkYjZmYWM0NDdhYTU1MjI2MDViZWYw&__proxy_form=0)
![{\displaystyle \int x^{3}r\,dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84ODg2YzlkZjI5ZjdjYjhhOWZiZTVkYzY5NGI3NGIzZjE1OTBkYWZi&__proxy_form=0)
![{\displaystyle \int x^{3}r^{3}\,dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNWNjZWIxYTNmNjk4YTM4YjAxMjA2OWU5ZThkNmJlMzJlZjUyMGYx&__proxy_form=0)
![{\displaystyle \int x^{3}r^{2n+1}\,dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{2}r^{2n+3}}{2n+3}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jMzc1NzhhYTEzMDlkNmFlMWIwNzJhNTBlYjlkZWEwMjUxYTA1OThh&__proxy_form=0)
![{\displaystyle \int x^{4}r\,dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left(x+r\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jZjc5YTIzYjA3ZTJiNjIwZTc3YTkyMTM0NDEyNzQwODU1MWMwZmJl&__proxy_form=0)
![{\displaystyle \int x^{4}r^{3}\,dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left(x+r\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mZmM0OTcwMzJkN2NiMzRlNWFiMDFhMjI3YjU3NDdjYmY0OTEwYTc2&__proxy_form=0)
![{\displaystyle \int x^{5}r\,dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kMTU0NzQxZmNhNTcwOGNiZDdkMDYyZmVkMzM1YTY5YjhhNWM3YzVl&__proxy_form=0)
![{\displaystyle \int x^{5}r^{3}\,dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNGNiMTQ4ZGU0ZTMzNDA4NWNkYmUxOGNlNGY0NTY1OTkyNTA4MmY4&__proxy_form=0)
![{\displaystyle \int x^{5}r^{2n+1}\,dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNWE1ZGRiNzVkNDFiNTRjMWFiMTc0MzMzZDdjNmFlYTMyMWY5NGQ4&__proxy_form=0)
![{\displaystyle \int {\frac {r\,dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\,\operatorname {arsinh} {\frac {a}{x}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81YTUxMWI1NWIyNjI0YzM4MzI1NjUzZDRmNmU5YjI5MmZhZTJkYTQ5&__proxy_form=0)
![{\displaystyle \int {\frac {r^{3}\,dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80NTA4ZjhkN2VjYjczMzZkMTUzNDI1ZGVkNDBhY2U2Nzk4ODM0N2M5&__proxy_form=0)
![{\displaystyle \int {\frac {r^{5}\,dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wNWNhMWQ4OTY5NTA1OTU5MTZlYjEzOTkxYzVkYTg1NDI3MTQ1MjE3&__proxy_form=0)
![{\displaystyle \int {\frac {r^{7}\,dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lN2YwN2U0Zjk2ZDE2N2Q2ZDU0ZWJmNTZhOTIwOTgzM2JjNGJiNWEz&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81NGE3OGFhZDkyZjFmZDA5ZGY0ZjRjMzNkNWIyODA4MWFlYzA5YzQ1&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zZTkwM2FhY2JmZDgyOTE4NGIzZWViOTIzM2NlMTRhYzIzNmQxZTZi&__proxy_form=0)
![{\displaystyle \int {\frac {x\,dx}{r}}=r}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83N2EzMTU2YTMzODM0YTE5ZTE5MmFmNjdmM2E1NGRhNmRkYmU0Y2Ux&__proxy_form=0)
![{\displaystyle \int {\frac {x\,dx}{r^{3}}}=-{\frac {1}{r}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84ZWZjOGExNWYzMDcyNTU1MjMzYjI3ODFkYzRjMzk4NzgwZjkwNGU2&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\operatorname {arsinh} {\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left({\frac {x+r}{a}}\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mODExZTMzMDY3ZjhkNzc5ZDdhZTVkZmEzYTg0MmU3ZTY5YTFiMzY2&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83MjE3Zjc2NzQyNmM4ZmZiMmQyNTY3NGNmMDhkOGQ5YzFhMzAwMzVl&__proxy_form=0)
Integrals involving s = √x2 − a2
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Assume x2 > a2 (for x2 < a2, see next section):
![{\displaystyle \int s\,dx={\frac {1}{2}}\left(xs-a^{2}\ln \left|x+s\right|\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zNTg4NjQ0MmQ4YTQxNzE5NWMwNzBkZTA5ZTk0N2JkMTEwNDZmNmE3&__proxy_form=0)
![{\displaystyle \int xs\,dx={\frac {1}{3}}s^{3}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83OWVjNTA3MjQ5NTk1NDdkN2ExODAyOTZlZDgyMjMyM2FjMGRlYTlh&__proxy_form=0)
![{\displaystyle \int {\frac {s\,dx}{x}}=s-|a|\arccos \left|{\frac {a}{x}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82MTc5NWNkZmJlNmVkNDk1MzI3MGVhY2JjOTIzZjRjNjBiZGE2ZGI1&__proxy_form=0)
Here
where the positive value of
is to be taken.
![{\displaystyle \int {\frac {dx}{xs}}={\frac {1}{a}}\operatorname {arcsec} \left|{\frac {x}{a}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lMDFmZDdiYjc3MWRjZWNhNGYwZDcxM2NmMTNmYmI3ZTVkYjFlZWJm&__proxy_form=0)
![{\displaystyle \int {\frac {x\,dx}{s}}=s}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83NDk4ZTI0ZjViODEwMzM1YzA3NTIyYzBhOTczOTM1OGVmYzE3MWNl&__proxy_form=0)
![{\displaystyle \int {\frac {x\,dx}{s^{3}}}=-{\frac {1}{s}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80MjBlZTUyN2NjNjE5ZWFlYTFmODk3YjJkOGQ5NzY1ZTJhYzNhMzRh&__proxy_form=0)
![{\displaystyle \int {\frac {x\,dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kYTI4OTlmODEzMDkxYWI4NjhiMTM0ZGJiMzk4NjUxZmQ2N2NkZDRh&__proxy_form=0)
![{\displaystyle \int {\frac {x\,dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81MzRiMWUwZjUyNDhkZmQzMzE5MzE3M2QyODBhNmFiMGFmYTczZDMw&__proxy_form=0)
![{\displaystyle \int {\frac {x\,dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kZjIzYmUwOGRkYTBiMzVlZWVjNzkzYzgwYjQ0ZWY1ZGIwMzBkYzMw&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2m}\,dx}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\,dx}{s^{2n-1}}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lMTQ2ZWRmZDU2NmU0NDQ4ZDFkMTJkNWJkNmFiZWZmMTcxMDE5NjE5&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lMzk4NGUzNTMzMTZmZDA4NDVkN2UyYTUyNWEzYzIwY2RmM2IzZmVm&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xOTQyOWUwYWNjMTc3N2ZkZGU4MjBlOWZiZjg2MzkwMGMzYzNlYjE3&__proxy_form=0)
![{\displaystyle \int {\frac {x^{4}\,dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNTEzOWM0MWRlNjU4MWQ4OGI1MjcxZjZmNDgxM2NhYWZhZjRkZDFh&__proxy_form=0)
![{\displaystyle \int {\frac {x^{4}\,dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNTRhN2I0NTU2MzZhMTA1OWNmMTU4NzRiMjlmYjU2NGU1MjhiNWRl&__proxy_form=0)
![{\displaystyle \int {\frac {x^{4}\,dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wY2M1OTkzMTVjOWQzYzFjMWQyYTBjZjU4MmMxZTg5YzNlZmM5M2Iz&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2m}\,dx}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad {\mbox{(}}n>m\geq 0{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iMWI0ZTc2NjhlMDk4NjJmZTFiOTI4ODE3NjcyNDdiNTc3Y2YyZDg1&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82NzZhYzIzMDhlZDYwMjE4ZjRlMjQ2ODg0ZTU3ODNkZjhlMmViYzU0&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wNTRhNTk1OWNlNWUwM2NmMjc5YzFiMjlkZmYyYmEwMTRhYzZkY2Rl&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84Njg0MzMxMWRlN2ZjNzJiYzAxZjg3NzQyNDQ1ZjdjNGI4ODg5OWU5&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jYTMyYjNhOGQ3ZjkwNDA4NDBmNWQxZGUzNDY3MTI5ZWRmZjBkODBi&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNzY3OGRkYmNjNzQ0OTNkY2VmMjY3NTMxYzQxMWRlNzUwZTZlY2Zj&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82YTk4MDU3Y2YzZjNkNmI3MDI1MTE0NDQ1Yzk3MmJiNmI3YjdhZjlk&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85Y2NlNGI4N2U3YTQ3Y2U0MjA0MjgwMzAzODEzOWY4MzBhZmQ1ZDM3&__proxy_form=0)
Integrals involving u = √a2 − x2
[edit]
![{\displaystyle \int u\,dx={\frac {1}{2}}\left(xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mYmY5MDhiN2RjMmVkNmEwZTBlMTc2M2VjZDU0MThiMmYzNzRiMWI4&__proxy_form=0)
![{\displaystyle \int xu\,dx=-{\frac {1}{3}}u^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81OWRiMTNkMDMyODc5NTlhZmE5NDdlMTQ5NmI3N2JlOTIzZWI4ODAz&__proxy_form=0)
![{\displaystyle \int x^{2}u\,dx=-{\frac {x}{4}}u^{3}+{\frac {a^{2}}{8}}(xu+a^{2}\arcsin {\frac {x}{a}})\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iNTc2OGUyMTYzN2IwNzYwODA4NDg2YWIwMDhjZTI4ZTM3ZGRjYjFh&__proxy_form=0)
![{\displaystyle \int {\frac {u\,dx}{x}}=u-a\ln \left|{\frac {a+u}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wODk4OWE2ZDNjZGE4NTA5NDg3NjI0MmZhYmU2M2VlZWNjOTcwNGI0&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{u}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kYWNlZmI4ZmI0ZGY0YTRjZDkwYjdlMjQ3MDZiMzljZTUzYTY2MDIz&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{u}}={\frac {1}{2}}\left(-xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85ODlkODU2M2RkOTE5NTg3OGZmMDJkNzJhYzgyODYxYTljNDhlMjg2&__proxy_form=0)
![{\displaystyle \int u\,dx={\frac {1}{2}}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(for }}|x|\geq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yODQ1Y2U1ZGU1ZjUwN2U2OTdmZDZiMDg0M2QyMDAwYWRhNTk4YjI0&__proxy_form=0)
![{\displaystyle \int {\frac {x}{u}}\,dx=-u\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82NDFkNWQ4NWEyOTUyNTM1NWRmY2YzN2ZlNDU0NjA1NDMwMGNjMDRk&__proxy_form=0)
Integrals involving R = √ax2 + bx + c
[edit]
Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad {\mbox{(for }}a>0{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zYWRkMmVhNmQwNDY1ZGMxZjhmM2RlNzE5MzEwNGY4YmFkNWI3YTRi&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84MzdlMWFiOTFmZTg5OWU4OGI2ZjNjNGIxMzY2NmU4Njk3ZWIzMDEz&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81NTZlZmRjYmY4ZWU5MmJmYjI4ZTQ4NDgyMTQ5Yzc2OWQ5MTI1MDUy&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{, }}\left|2ax+b\right|<{\sqrt {b^{2}-4ac}}{\mbox{)}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81YjQwMTdhNjBhOGVkYjUwNWZlMjE0OWE3NzJkNWUyMzFjMWYxZWQ5&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wODY5MzRiMjk0ZThiNTNiZWJlN2I1MzI0MWJhZDkxMmY0MjEyZGVl&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82ODg3ZWZmNTVlNDRhZjdlZDAzMWZhMWQ5MTlkM2RlM2YzNzlhOTBi&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{R^{2n+1}}}={\frac {2}{(2n-1)(4ac-b^{2})}}\left({\frac {2ax+b}{R^{2n-1}}}+4a(n-1)\int {\frac {dx}{R^{2n-1}}}\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hNmZkMTljODJhYmZkNmFiMDFkOTNjYzNmMjY5MTA1OWQ0YjQ5MTVj&__proxy_form=0)
![{\displaystyle \int {\frac {x}{R}}\,dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lNjk3OGViNzZkNGU3NDM1ZjIyN2EzNmQ3NTI4NGZkNWFjMzY2YTc2&__proxy_form=0)
![{\displaystyle \int {\frac {x}{R^{3}}}\,dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mOTM2NDA5NmRkMWMwNjIxNjQ2YWVmZWYyNzA1OTQ1MjQwYTUyNjUw&__proxy_form=0)
![{\displaystyle \int {\frac {x}{R^{2n+1}}}\,dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xMTYwYmI4MmM5MDg3MWI3NTg3NzI4Nzc3ZWUwM2Y5YzU5OTUzNzU3&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left|{\frac {2{\sqrt {c}}R+bx+2c}{x}}\right|,~c>0}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lNjJmOGQyZDJlZGRhNWY2MzhlYTQwZDY5ZGE3ZDliNGVlMjBkYmZl&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right),~c<0}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85NGU0NGEzYjhhMmJhYWVjZWZlMTQyNmFiZTlmNjZiNDgzYmFlODJk&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{xR}}={\frac {1}{\sqrt {-c}}}\operatorname {arcsin} \left({\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac}}}}\right),~c<0,b^{2}-4ac>0}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yMTQwZGI3MTBhNmYwOWI0YmM2Y2QyZjgxZDc5YjdjNDRkMzI1Nzkw&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {2}{bx}}\left({\sqrt {ax^{2}+bx}}\right),~c=0}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jMDEyMWVkMjcyMjQzYzRiMjE2OWVmZGYzOWZjYmYwZWE2ZmNmMWVk&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}}{R}}\,dx={\frac {2ax-3b}{4a^{2}}}R+{\frac {3b^{2}-4ac}{8a^{2}}}\int {\frac {dx}{R}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82ZmVkOWU1YTdhYjcxNTYzZWIwZmJhYWRjYjlkMGM5MmMwMDc4ZWFm&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{x^{2}R}}=-{\frac {R}{cx}}-{\frac {b}{2c}}\int {\frac {dx}{xR}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jN2VkMDkzZTkxMGIzZTM4NGI0NmU2MTdlYjE5ZDQ4OTU4MzRlMmYz&__proxy_form=0)
![{\displaystyle \int R\,dx={\frac {2ax+b}{4a}}R+{\frac {4ac-b^{2}}{8a}}\int {\frac {dx}{R}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83Y2E4NmU4ZTA4YWJmNzAzMGFjZDdkMDJlMWZkMWNmMTE0MjQyMTIw&__proxy_form=0)
![{\displaystyle \int xR\,dx={\frac {R^{3}}{3a}}-{\frac {b(2ax+b)}{8a^{2}}}R-{\frac {b(4ac-b^{2})}{16a^{2}}}\int {\frac {dx}{R}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80Y2RkYWQ1ODNiZjQ2ZTAyOGFhOTRjNzliNWVhMDQxZmIzNzMxOWFk&__proxy_form=0)
![{\displaystyle \int x^{2}R\,dx={\frac {6ax-5b}{24a^{2}}}R^{3}+{\frac {5b^{2}-4ac}{16a^{2}}}\int R\,dx}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85OGE3ZWYxZDM4YTc5MTM1ZjQzYzBiZWM4Yjk5NGQ5YTlmNmI2ZWMz&__proxy_form=0)
![{\displaystyle \int {\frac {R}{x}}\,dx=R+{\frac {b}{2}}\int {\frac {dx}{R}}+c\int {\frac {dx}{xR}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yNGYzNWNkNTFjMThlMGE0Yzg3NDI0YTJiNmMzMDMzMmE0OTJkN2U3&__proxy_form=0)
![{\displaystyle \int {\frac {R}{x^{2}}}\,dx=-{\frac {R}{x}}+a\int {\frac {dx}{R}}+{\frac {b}{2}}\int {\frac {dx}{xR}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83MGIyYzMxYzUxMjRlNjI3YTJmN2UxOWExZWQyM2NkNDEyYjg1YmQ2&__proxy_form=0)
![{\displaystyle \int {\frac {x^{2}\,dx}{R^{3}}}={\frac {(2b^{2}-4ac)x+2bc}{a(4ac-b^{2})R}}+{\frac {1}{a}}\int {\frac {dx}{R}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82YmQxZDZkZDU3MGM2ZDkzNjVkZWJkMjkzOGUyOWExYTY2NDZmNjk2&__proxy_form=0)
Integrals involving S = √ax + b
[edit]
![{\displaystyle \int S\,dx={\frac {2S^{3}}{3a}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jZWUxMjNkZTZmMTQ0ZWRkMjQzYmEyYjIzOWQzOTg2NzAwMTgzZDcy&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{S}}={\frac {2S}{a}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82NDU4YzhhNDJjOTMyMWE5ODRmM2VmYzY0YjkwNDVhYmQ3MDZjYTcw&__proxy_form=0)
![{\displaystyle \int {\frac {dx}{xS}}={\begin{cases}-{\dfrac {2}{\sqrt {b}}}\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\-{\dfrac {2}{\sqrt {b}}}\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\{\dfrac {2}{\sqrt {-b}}}\arctan \left({\dfrac {S}{\sqrt {-b}}}\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xNDQ5Zjk2YjNjOGNmYTdiMmFlMDc1NTQxYzBlYjA3NmUxMjJjNWY2&__proxy_form=0)
![{\displaystyle \int {\frac {S}{x}}\,dx={\begin{cases}2\left(S-{\sqrt {b}}\,\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\2\left(S-{\sqrt {b}}\,\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\2\left(S-{\sqrt {-b}}\arctan \left({\dfrac {S}{\sqrt {-b}}}\right)\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lODkwYzcyOGZmNmY5Y2I5MzRhY2M3NGJiODg0NjUwYzRkYmI5OGJl&__proxy_form=0)
![{\displaystyle \int {\frac {x^{n}}{S}}\,dx={\frac {2}{a(2n+1)}}\left(x^{n}S-bn\int {\frac {x^{n-1}}{S}}\,dx\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yMjBmYTg5NDc0ZDhjZDNmYjViMjllYWFlMDA4YjVlNjMwNDExYTQw&__proxy_form=0)
![{\displaystyle \int x^{n}S\,dx={\frac {2}{a(2n+3)}}\left(x^{n}S^{3}-nb\int x^{n-1}S\,dx\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lMzUyYWU5NWM2Mzk2OWZiMjAzYTI5OTgwZjQzNzA4NjkxY2RhZTI5&__proxy_form=0)
![{\displaystyle \int {\frac {1}{x^{n}S}}\,dx=-{\frac {1}{b(n-1)}}\left({\frac {S}{x^{n-1}}}+\left(n-{\frac {3}{2}}\right)a\int {\frac {dx}{x^{n-1}S}}\right)}](./?url=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kYzJiZmE4OTVhNGQ4NGUxMjZkOTk3YzFmNzZkMDlmMDcxMmEzMjFh&__proxy_form=0)
- Abramowitz, Milton; Stegun, Irene A., eds. (1972). "Chapter 3". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover.
- Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276. (Several previous editions as well.)
- Peirce, Benjamin Osgood (1929) [1899]. "Chapter 3". A Short Table of Integrals (3rd revised ed.). Boston: Ginn and Co. pp. 16–30.