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| <math> \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x^3}dx = - \dfrac{\left(x^2+a^2\right)^{3/2}}{2x^2} + \dfrac{3}{2}\sqrt{x^2+a^2} - \dfrac{3}{2}a\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C</math> | | <math> \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x^3}dx = - \dfrac{\left(x^2+a^2\right)^{3/2}}{2x^2} + \dfrac{3}{2}\sqrt{x^2+a^2} - \dfrac{3}{2}a\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C</math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | <math> x^3+a^3</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x^3+a^3} = \dfrac{1}{6a^2} \ln \dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{1}{a^2\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} +C</math> | ||
| + | |- | ||
| + | | <math> \dfrac{xdx}{x^3+a^3} = \dfrac{1}{6a}\ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{a\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}}+C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{x^2dx}{x^3+a^3} = \dfrac{1}{3} \ln\left(x^3+a^3\right)+C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3} \ln \left(\dfrac{x^3}{x^3+a^3}\right)+C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^3x} - \dfrac{1}{6a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)} - \dfrac{1}{a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}}+C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{\left(x^3+a^3\right)^2} = \dfrac{x}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{9a^5} \ln\dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{2}{3a^5\sqrt{3}}\arctan\dfrac{2x-a}{a\sqrt{3}} +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{xdx}{\left(x^3+a^3\right)^2} = \dfrac{x^2}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{18a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{3a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{x^2dx}{\left(x^3+a^3\right)} = -\dfrac{1}{3\left(x^3+a^3\right)} +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{3a^6}\ln\left(\dfrac{x^3}{x^3+a^3}\right) </math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^6x} - \dfrac{x^2}{3a^6\left(x^3+a^3\right)} - \dfrac{4}{3a^6}\int \dfrac{xdx}{x^3+a^3} </math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{x^{m}dx}{x^3+a^3} = \dfrac{x^{m-2}}{m-2} - a^3 \int \dfrac{x^{m-3}dx}{x^3+a^3} </math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x^{n}\left(x^3+a^3\right)} = \dfrac{-1}{a^3\left(n-1\right)x^{n-1}} - \dfrac{1}{a^3} \int \dfrac{dx}{x^{n-3}\left(x^3+a^3\right)} </math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | <math> x^4 + a^4</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x^4+a^4} = \dfrac{1}{4a^3\sqrt{2}}\ln\left(\dfrac{x^2+ax\sqrt{2}+a^2}{x^2-ax\sqrt{2}+a^2}\right) - \dfrac{1}{2a^3\sqrt{2}}\arctan \dfrac{ax\sqrt{2}}{x^2-a^2} +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{xdx}{x^4+a^4} = \dfrac{1}{2a^2} \arctan \dfrac{x^2}{a^2} +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{x^2da}{x^4+a^4} = \dfrac{1}{4a\sqrt{2}} \ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) = \dfrac{1}{2a\sqrt{2}} \arctan \dfrac{ax\sqrt{2}}{x^2-a^2} +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{x^3dx}{x^4+a^4} = \dfrac{1}{4} \ln \left(x^4+a^4\right) +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x\left(x^4+a^4\right)} = \dfrac{1}{4a^4}\ln\left(\dfrac{x^4}{x^4+a^4}\right) +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x^2\left(x^4+a^4\right)} = -\dfrac{1}{a^4x} - \dfrac{1}{4a^5\sqrt{2}}\ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) + \dfrac{1}{2a^5\sqrt{2}}\arctan\dfrac{ax\sqrt{2}}{x^2-a^2} +C</math> | ||
| + | |- | ||
| + | | <math> \int \dfrac{dx}{x^3\left(x^4+a^4\right)} = -\dfrac{1}{2a^4x^2} - \dfrac{1}{a^6} \arctan\dfrac{x^2}{a^2} +C</math> | ||
|} | |} | ||
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Revision as of 16:59, 26 February 2015
Indefinite Integrals with $ x^n+a^n $
click here for more formulas
| $ x^2 + a^2 $ | |
|---|---|
| $ \int \frac {d x}{ x^2 + a^2} \ = \ \frac {1}{a} \arctan \ \frac {x}{a} +C $ | |
| $ \int \frac {x \ d x}{x^2 + a^2} = \frac {1}{2} \ln \left ( x^2 + a^2 \right ) +C $ | |
| $ \int \frac {x^2 \ d x }{x^2 + a^2} = x \ - \ a \arctan \frac {x}{a} +C $ | |
| $ \int \frac {x^3 \ d x}{x^2 + a^2} = \frac{x^2}{2} - \frac{a^2}{2} \ln (x^2+a^2) +C $ | |
| $ \int \frac {d x}{ x(x^2 + a^2)} = \frac {1}{2a^2} \ln \left ( \frac {x^2}{x^2 + a^2} \right )+C $ | |
| $ \int \frac {d x}{x^2(x^2 + a^2)} = - \frac {1}{a^2x} - \frac {1}{a^3} \arctan \frac {x}{a}+C $ | |
| $ \int \frac {d x}{x^3(x^2 + a^2)} = - \frac {1}{2a^2x^2} - \frac {1}{2a^4} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) +C $ | |
| $ \int \frac {d x}{(x^2 +a^2)^2} =\frac {x}{2a^2(x^2+a^2)} + \frac {1}{2a^3} \arctan \frac {x}{a} +C $ | |
| $ \int \frac {x d x}{(x^2 +a^2)^2} = \frac {-1}{2(x^2 + a^2)}+C $ | |
| $ \int \frac {x^2 d x}{(x^2 +a^2)^2} = \frac {-x}{2(x^2 + a^2)} + \frac {1}{2a} \arctan \frac {x}{a}+C $ | |
| $ \int \frac {x^3 d x}{(x^2 +a^2)^2} = \frac {a^2}{2(x^2 + a^2)} + \frac{1}{2} \ln(x^2 + a^2) +C $ | |
| $ \int \frac {d x}{ x(x^2 + a^2)^2} = \frac {1}{2a^2(x^2+a^2)} + \frac{1}{2a^4} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) +C $ | |
| $ \int \frac {d x}{x^2(x^2 + a^2)^2} = - \frac {1}{a^4x} - \frac {x}{2a^4(x^2 + a^2)} - \frac {3}{2a^5} \arctan \frac {x}{a} +C $ | |
| $ \int \frac {d x}{ x^3(x62 +a^2)^2} = - \frac {1}{2a^4x^2} - \frac {1}{2a^4(x^2+a^2)} - \frac {1}{a^6} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) +C $ | |
| $ \int \frac {d x}{(x^2 + a^2)^n} = \frac {x}{2(n-1)a^2(x^2 + a^2)^{n-1}} + \frac {2n -3}{(2n - 2)a^2} \int \frac {d x}{(x^2 + a^2)^{n-1}} +C $ | |
| $ \int \frac {x dx}{(x^2+a^2)^n} = \frac {-1}{2(n-1)(x^2 + a^2)^{n-1}} +C $ | |
| $ \int \frac {d x}{x(x^2 +a^2)^n} = \frac {1}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac {1}{a^2} \int \frac {d x}{x(x^2 + a^2)^{n-1}} +C $ | |
| $ \int \frac {x^m d x}{(x^2 + a^2)^n} = \int \frac {x^{m-2} d x}{(x^2+a^2)^{n-1}} - a^2 \ \int \frac {x^{m-2} d x}{(x^2 + a^2)^n}+C $ | |
| $ \int \frac {d x}{x^m (x^2 +a^2)^n} = \frac {1}{a^2} \int \frac {d x}{x^m(x^2+a^2)^{n-1}} - \frac {1}{a^2} \int \frac {d x}{ x^{m-2}(x^2+a^2)^n} +C $ | |
| $ \sqrt{x^2+a^2} $ | |
| $ \int \dfrac{dx}{\sqrt{x^2+a^2}} = \ln\left(x+\sqrt{x^2+a^2}\right) \qquad o\grave{u}\qquad Arg sh \dfrac{x}{a}+C $ | |
| $ \int \dfrac{xdx}{\sqrt{x^2+a^2}} = \sqrt{x^2+a^2}+C $ | |
| $ \int \dfrac{x^2dx}{\sqrt{x^2+a^2}} = \dfrac{x\sqrt{x^2+a^2}}{2} - \dfrac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int \dfrac{x^3dx}{\sqrt{x^2+a^2}} = \dfrac{\left(x^2+a^2\right)^{3/2}}{3} - a^2\sqrt{x^2+a^2}+C $ | |
| $ \int \dfrac{dx}{x\sqrt{x^2+a^2}} = -\dfrac{1}{a} \ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ \int \dfrac{dx}{x^2\sqrt{x^2+a^2}} = -\dfrac{\sqrt{x^2+a^2}}{a^2x}+C $ | |
| $ \int \dfrac{dx}{x^3\sqrt{x^2+a^2}} = -\dfrac{\sqrt{x^2+a^2}}{2a^2x^2} + \dfrac{1}{2a^3}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ \int \sqrt{x^2+a^2}dx = \dfrac{x\sqrt{x^2+a^2}}{2} + \dfrac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int x\sqrt{x^2+a^2}dx = \dfrac{\left(x^2+a^2\right)^{3/2}}{3}+C $ | |
| $ \int x^2\sqrt{x^2+a^2}dx = \dfrac{x\left(x^2+a^2\right)^{3/2}}{4} - \dfrac{a^2x\sqrt{x^2+a^2}}{8} - \dfrac{a^4}{8}\ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int x^3\sqrt{x^2+a^2}dx = \dfrac{\left(x^2+a^2\right)^{5/2}}{5} - \dfrac{a^2\left(x^2+a^2\right)^{3/2}}{3}+C $ | |
| $ \int \dfrac{\sqrt{x^2+a^2}}{x}dx = \sqrt{x^2+a^2} - a\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ \int \dfrac{\sqrt{x^2+a^2}}{x^2}dx = -\dfrac{\sqrt{x^2+a^2}}{x} + \ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int \dfrac{\sqrt{x^2+a^2}}{x^3}dx = -\dfrac{\sqrt{x^2+a^2}}{2x^2} - \dfrac{1}{2a}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ \int \dfrac{dx}{\left(x^2+a^2\right)^{3/2}} = \dfrac{x}{a^2\sqrt{x^2+a^2}}+C $ | |
| $ \int \dfrac{xdx}{\left(x^2+a^2\right)^{3/2}} = \dfrac{-1}{\sqrt{x^2+a^2}}+C $ | |
| $ \int \dfrac{x^2dx}{\left(x^2+a^2\right)^{3/2}} = \dfrac{-x}{\sqrt{x^2+a^2}} +\ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int \dfrac{x^3dx}{\left(x^2+a^2\right)^{3/2}} = \sqrt{x^2+a^2} + \dfrac{a^2}{\sqrt{x^2+a^2}}+C $ | |
| $ \int \dfrac{dx}{x\left(x^2+a^2\right)^{3/2}} = \dfrac{1}{a^2\sqrt{x^2+a^2}} - \dfrac{1}{a^3}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ \int \dfrac{dx}{x^2\left(x^2+a^2\right)^{3/2}} = -\dfrac{\sqrt{x^2+a^2}}{a^4x} - \dfrac{x}{a^4\sqrt{x^2+a^2}}{x}+C $ | |
| $ \int \dfrac{dx}{x^3\left(x^2+a^2\right)^{3/2}} = \dfrac{-1}{2a^2x^2\sqrt{x^2+a^2}} - \dfrac{3}{2a^4\sqrt{x^2+a^2}} + \dfrac{3}{2a^5}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ \int \left(x^2+a^2\right)^{3/2}dx = \dfrac{x\left(x^2+a^2\right)^{3/2}}{4} + \dfrac{3a^2x\sqrt{x^2+a^2}}{8} + \dfrac{3}{8}a^4\ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int x\left(x^2+a^2\right)^{3/2}dx = \dfrac{\left(x^2+a^2\right)^{5/2}}{5}+C $ | |
| $ \int x^2\left(x^2+a^2\right)^{3/2}dx = \dfrac{x\left(x^2+a^2\right)^{5/2}}{6} - \dfrac{a^2x\left(x^2+a^2\right)^{3/2}}{24} - \dfrac{a^4x\sqrt{x^2+a^2}}{16} + \dfrac{a^6}{16}\ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int x^3\left(x^2+a^2\right)^{3/2}dx = \dfrac{\left(x^2+a^2\right)^{7/2}}{7} - \dfrac{a^2\left(x^2+a^2\right)^{5/2}}{5}+C $ | |
| $ \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x}dx = \dfrac{\left(x^2+a^2\right)^{3/2}}{3} + a^2\sqrt{x^2+a^2} - a^3\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x^2}dx = - \dfrac{\left(x^2+a^2\right)^{3/2}}{x} + \dfrac{3x\sqrt{x^2+a^2}}{2} + \dfrac{3}{2}a^2\ln\left(x+\sqrt{x^2+a^2}\right)+C $ | |
| $ \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x^3}dx = - \dfrac{\left(x^2+a^2\right)^{3/2}}{2x^2} + \dfrac{3}{2}\sqrt{x^2+a^2} - \dfrac{3}{2}a\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right)+C $ | |
| $ x^3+a^3 $ | |
| $ \int \dfrac{dx}{x^3+a^3} = \dfrac{1}{6a^2} \ln \dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{1}{a^2\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} +C $ | |
| $ \dfrac{xdx}{x^3+a^3} = \dfrac{1}{6a}\ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{a\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}}+C $ | |
| $ \int \dfrac{x^2dx}{x^3+a^3} = \dfrac{1}{3} \ln\left(x^3+a^3\right)+C $ | |
| $ \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3} \ln \left(\dfrac{x^3}{x^3+a^3}\right)+C $ | |
| $ \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^3x} - \dfrac{1}{6a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)} - \dfrac{1}{a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}}+C $ | |
| $ \int \dfrac{dx}{\left(x^3+a^3\right)^2} = \dfrac{x}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{9a^5} \ln\dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{2}{3a^5\sqrt{3}}\arctan\dfrac{2x-a}{a\sqrt{3}} +C $ | |
| $ \int \dfrac{xdx}{\left(x^3+a^3\right)^2} = \dfrac{x^2}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{18a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{3a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} +C $ | |
| $ \int \dfrac{x^2dx}{\left(x^3+a^3\right)} = -\dfrac{1}{3\left(x^3+a^3\right)} +C $ | |
| $ \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{3a^6}\ln\left(\dfrac{x^3}{x^3+a^3}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^6x} - \dfrac{x^2}{3a^6\left(x^3+a^3\right)} - \dfrac{4}{3a^6}\int \dfrac{xdx}{x^3+a^3} $ | |
| $ \int \dfrac{x^{m}dx}{x^3+a^3} = \dfrac{x^{m-2}}{m-2} - a^3 \int \dfrac{x^{m-3}dx}{x^3+a^3} $ | |
| $ \int \dfrac{dx}{x^{n}\left(x^3+a^3\right)} = \dfrac{-1}{a^3\left(n-1\right)x^{n-1}} - \dfrac{1}{a^3} \int \dfrac{dx}{x^{n-3}\left(x^3+a^3\right)} $ | |
| $ x^4 + a^4 $ | |
| $ \int \dfrac{dx}{x^4+a^4} = \dfrac{1}{4a^3\sqrt{2}}\ln\left(\dfrac{x^2+ax\sqrt{2}+a^2}{x^2-ax\sqrt{2}+a^2}\right) - \dfrac{1}{2a^3\sqrt{2}}\arctan \dfrac{ax\sqrt{2}}{x^2-a^2} +C $ | |
| $ \int \dfrac{xdx}{x^4+a^4} = \dfrac{1}{2a^2} \arctan \dfrac{x^2}{a^2} +C $ | |
| $ \int \dfrac{x^2da}{x^4+a^4} = \dfrac{1}{4a\sqrt{2}} \ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) = \dfrac{1}{2a\sqrt{2}} \arctan \dfrac{ax\sqrt{2}}{x^2-a^2} +C $ | |
| $ \int \dfrac{x^3dx}{x^4+a^4} = \dfrac{1}{4} \ln \left(x^4+a^4\right) +C $ | |
| $ \int \dfrac{dx}{x\left(x^4+a^4\right)} = \dfrac{1}{4a^4}\ln\left(\dfrac{x^4}{x^4+a^4}\right) +C $ | |
| $ \int \dfrac{dx}{x^2\left(x^4+a^4\right)} = -\dfrac{1}{a^4x} - \dfrac{1}{4a^5\sqrt{2}}\ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) + \dfrac{1}{2a^5\sqrt{2}}\arctan\dfrac{ax\sqrt{2}}{x^2-a^2} +C $ | |
| $ \int \dfrac{dx}{x^3\left(x^4+a^4\right)} = -\dfrac{1}{2a^4x^2} - \dfrac{1}{a^6} \arctan\dfrac{x^2}{a^2} +C $ | |
