Indefinite Integrals with $ x^n+a^n $
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| $ 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 $ | |
| $ x^{n} + a^{n} $ | |
| $ \int \dfrac{dx}{x\left(x^n+a^n\right)} = \dfrac{1}{na^{n}}\ln\dfrac{x^{n}}{x^{n}+a^{n}} +C $ | |
| $ \int \dfrac{x^{n-1}}{x^{n}+a^{n}} = \dfrac{1}{n}\ln\left(x^{n}+a^{n}\right) +C $ | |
| $ \int \dfrac{x^{m}}{\left(x^{n}+a^{n}\right)^{r}} = \int \dfrac{x^{m-n}dx}{\left(x^{n}+a^{n}\right)^{r-1}} - a^{n}\int\dfrac{x^{m-n}dx}{\left(x^{n}+a^{n}\right)^{r}} +C $ | |
| $ \int \dfrac{dx}{x^{m}\left(x^{n}+a^{n}\right)^{r}} = \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m}\left(x^{n}+a{n}\right)^{r-1}} - \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m-n}\left(x^{n}+a^{n}\right)^{r}} +C $ | |
| $ \int \dfrac{dx}{x\sqrt{x^{n}+a^{n}}} = \dfrac{1}{n\sqrt{a^{n}}}\ln\left(\dfrac{\sqrt{x^{n}+a^{n}}-\sqrt{a^{n}}}{\sqrt{x^{n}+a^{n}}+\sqrt{a^{n}}}\right) +C $ | |
| $ \int \dfrac{dx}{x\left(x^{n}-a^{n}\right)} = \dfrac{1}{na^{n}}\ln\left(\dfrac{x^{n}-a^{n}}{x^{n}}\right) +C $ | |
| $ \int \dfrac{x^{n-1}dx}{x^{n}-a^{n}} = \dfrac{1}{n}\ln\left(x^{n}-a^{n}\right) +C $ | |
| $ \int \dfrac{x^{m}dx}{\left(x^{n}-a^{n}\right)^{r}} = a^{n}\int\dfrac{x^{m-n}dx}{\left(x^{n}-a^{n}\right)^{r}} + \int\dfrac{x^{m-n}dx}{\left(x^{n}-a^{n}\right)^{r-1}} $ | |
| $ \int\dfrac{dx}{x^{m}\left(x^{n}-a^{n}\right)^{r}} = \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m-n}\left(x^{n}-a^{n}\right)^{r}}-\dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m}\left(x^{n}-a^{n}\right)^{r-1}} $ | |
| $ \int\dfrac{dx}{x\sqrt{x^{n}-a^{n}}} = \dfrac{2}{n\sqrt{a^{n}}}\arccos\sqrt{\dfrac{a^{n}}{x^{n}}} $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m}+a^{2m}} = \dfrac{1}{ma^{2m-p}} \sum_{k=1}^m \sin\dfrac{\left(2k-1\right)p\pi}{2m}\arctan\left(\dfrac{x+a\cos\left[\left(2k-1\right)\pi /2m\right]}{a\sin\left[\left(2k-1\right)\pi /2m\right]}\right) - \dfrac{1}{2ma^{2m-p}} \sum_{k=1}^m \cos\dfrac{\left(2k-1\right)p\pi}{2m}\ln\left(x^2+2ax\cos\dfrac{\left(2k-1\right)\pi}{2m} + a^2\right) $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m}-a^{2m}} = \dfrac{1}{2ma^{2m-p}} \sum_{k=1}^{m-1} \cos\dfrac{kp\pi}{m}\ln\left(x^2 - 2ax\cos\dfrac{k\pi}{m} + a^2\right) - \dfrac{1}{ma^{2m-p}} \sum_{k=1}^{m-1} \sin \dfrac{kp\pi}{m} \arctan\left(\dfrac{x-a\cos\left(k\pi /m\right)}{a\sin\left(k\pi/m\right)}\right) + \dfrac{1}{2ma^{2m-p}}\left\{\ln\left(x-a\right)+\left(-1\right)^{p}\ln\left(x+a\right)\right\} $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m+1}+a^{2m+1}} = \dfrac{2\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m\sin\dfrac{2kp\pi}{2m+1}\arctan\left(\dfrac{x+a\cos\left[2k\pi/ \left(2m+1\right)\right]}{a\sin\left[2k\pi/ \left(2m+1\right)\right]}\right)-\dfrac{\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m \cos\dfrac{2kp\pi}{2m+1}\ln\left(x^2+2ax\cos\dfrac{2k\pi}{2m+1}+a^2\right) + \dfrac{\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}} $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m+1}-a^{2m+1}} = \dfrac{-2}{\left(2m+1\right)a^{2m-p+1}} \sum_{k=1}^m \sin \dfrac{2kp\pi}{2m+1}\arctan\left(\dfrac{x-a\cos\left[2k\pi/\left(2m+1\right)\right]}{a\sin\left[2k\pi/\left(2m+1\right)\right]}\right) + \dfrac{1}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m \cos\dfrac{2kp\pi}{2m+1}\ln\left(x^2-2ax\cos\dfrac{2k\pi}{2m+1}+a^2\right) + \dfrac{\ln\left(x-a\right)}{\left(2m+1\right)a^{2m-p+1}} $ | |
