Indefinite Integrals with $ x^n-a^n $
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| $ x^2 - a^2 , x> a $ | |
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| $ \int \frac {d x}{x^2 - a^2} = \frac {1}{2a} \ln \left ( \frac {x-a}{x+a} \right )+C \text{ or } -\frac{1}{a} \operatorname{argcoth}\,\frac {x}{a}+C $ | |
| $ \int \frac {x d x}{x^2 - a^2} = \frac {1}{2} \ln(x^2 - a^2)+C $ | |
| $ \int \frac {x^2 d x}{x^2 -a^2} = x + \frac{a}{2} \ln \left ( \frac {x-a}{x+a} \right )+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 -a^2}{x^2} \right )+C $ | |
| $ \int \frac {d x}{x^2(x^2-a^2)} = \frac {1}{a^2x} + \frac {1}{2a^3} \ln \left ( \frac {x-a}{x+a} \right ) +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}{4a^3} \ln \left ( \frac {x-a}{x+a} \right ) +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}{4a} \ln \left ( \frac {x-a}{x+a} \right ) +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}{4a^5} \ln \left ( \frac {x-a}{x+a} \right ) +C $ | |
| $ \int \frac {d x}{x^3(x^2 -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 d x}{(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}} $ | |
| $ \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} $ | |
| $ \int \frac {d x}{x^m (x^2 -a^2)^n} =\frac {1}{a^2} \int \frac {d x}{x^{m-2} (x^2 -a^2)^n} - \frac{1}{a^2} \ \int \frac {d x}{x^m (x^2 -a^2)^{n-1}} $ | |
| $ \sqrt{x^2-a^2} $ | |
| $ \int \dfrac{dx}{\sqrt{x^2-a^2}} = \ln\left(x+\sqrt{x^2-a^2}\right) $ or $ argcosh \dfrac{x}{a} $ | |
| $ \int \dfrac{xdx}{\sqrt{x^2-a^2}} = \sqrt{x^2-a^2} $ | |
| $ \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} \arccos \left|\frac{a}{x}\right \vert +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} \arccos \left|\frac{a}{x}\right \vert +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^2 x\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 \arccos \left|\frac{a}{x}\right \vert +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} \arccos \left|\frac{a}{x}\right \vert +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}{a^2\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}{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} \arccos \left|\frac{a}{x}\right \vert +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}} +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} \arccos \left|\frac{a}{x}\right \vert +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{x\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}a^4\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 \arccos \left | \dfrac{a}{x} \right \vert +C $ | |
| $ \int \dfrac{\left(x^2-a^2\right)^{3/2}}{x^2} dx = \dfrac{x\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{x\left(x^2-a^2\right)^{3/2}}{2x^2} + \dfrac{3\sqrt{x^2-a^2}}{2} - \dfrac{3}{2} a \arccos \left|\frac{a}{x}\right \vert +C $ | |
| $ \sqrt{a^2-x^2} $ | |
| $ \int \dfrac{dx}{\sqrt{a^2-x^2}} = \arcsin \dfrac{x}{a}+C $ | |
| $ \int \dfrac{xdx}{\sqrt{a^2-x^2}} = -\sqrt{a^2-x^2}+C $ | |
| $ \int \dfrac{x^2dx}{\sqrt{a^2-x^2}} = -\dfrac{x\sqrt{a^2-x^2}}{2} + \dfrac{a^2}{2} Arc sin \dfrac{x}{a}+C $ | |
| $ \int \dfrac{x^3 dx}{\sqrt{a^2-x^2}} = \dfrac{\left( a^2 - x^2 \right)^{3/2}}{3} - a^2 \sqrt{a^2-x^2} +C $ | |
| $ \int \dfrac{dx}{x\sqrt{a^2-x^2}} = -\dfrac{1}{a}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right)+C $ | |
| $ \int \dfrac{dx}{x^2\sqrt{a^2-x^2}} = -\dfrac{\sqrt{a^2-x^2}}{a^2x}+C $ | |
| $ \int \dfrac{dx}{x^3\sqrt{a^2-x^2}} = -\dfrac{\sqrt{a^2-x^2}}{2a^2x^2} -\dfrac{1}{2a^3}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right)+C $ | |
| $ \int \sqrt{a^2-x^2} dx = \dfrac{x\sqrt{a^2-x^2}}{2} - \dfrac{a^2}{2} \arcsin \dfrac{x}{a} +C $ | |
| $ \int x\sqrt{a^2-x^2} dx = -\dfrac{x\left(a^2-x^2\right)^{3/2}}{3} +C $ | |
| $ \int x^2\sqrt{a^2-x^2} dx =- \dfrac{x\left(a^2-x^2\right)^{3/2}}{4} + \dfrac{a^2x\sqrt{x^2-a^2}}{8} + \dfrac{a^4}{8} \arcsin \dfrac{a}{x}+C $ | |
| $ \int x^3 \sqrt{a^2-x^2} dx =\dfrac{x\left(a^2-x^2\right)^{5/2}}{5} - \dfrac{a^2\left(a^2-x^2\right)^{3/2}}{3}+C $ | |
| $ \int \dfrac{\sqrt{a^2 - x^2}}{x} dx = \sqrt{a^2-x^2} - a \ln \left( \dfrac{a+\sqrt{a^2-x^2}}{x} \right)+C $ | |
| $ \int \ dfrac{\sqrt{a^2-x^2}}{x^2} dx = - \dfrac{\sqrt{a^2-x^2}}{x} - \arcsin \dfrac{x}{a}+C $ | |
| $ \int \dfrac{\sqrt{a^2-x^2}}{x^3} dx = -\dfrac{\sqrt{a^2-x^2}}{2x^2} + \dfrac{1}{2a}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right)+C $ | |
| $ \int \dfrac{dx}{\left(a^2-x^2\right)^{3/2}} = \dfrac{x}{a^2\sqrt{a^2-x^2}} +C $ | |
| $ \int \dfrac{xdx}{\left(a^2-x^2\right)^{3/2}} = \dfrac{1}{\sqrt{a^2-x^2}} +C $ | |
| $ \int \dfrac{x^2 dx}{\left(a^2-x^2 \right)^{3/2}} = \dfrac{x}{\sqrt{a^2-x^2}} - \arcsin \dfrac{x}{a}+C $ | |
| $ \int \dfrac{x^3dx}{\left(a^2-x^2\right)^{3/2}} = \sqrt{a^2-x^2} + \dfrac{a^2}{\sqrt{a^2-x^2}} +C $ | |
| $ \int \dfrac{dx}{x\left(a^2-x^2\right)^{3/2}} = \dfrac{1}{a^2\sqrt{a^2-x^2}} - \dfrac{1}{a^3}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right)+C $ | |
| $ \int \dfrac{dx}{x^2\left(a^2-x^2\right)^{3/2}} = -\dfrac{\sqrt{a^2-x^2}}{a^4x} + \dfrac{x}{a^4\sqrt{a^2-x^2}}+C $ | |
| $ \int \dfrac{dx}{x^3\left(a^2-x^2\right)^{3/2}} = \dfrac{-1}{2a^2x^2\sqrt{a^2-x^2}} + \dfrac{3}{2a^4\sqrt{a^2-x^2}} - \dfrac{3}{2a^5}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right)+C $ | |
| $ \int \left(a^2-x^2\right)^{3/2} dx = \dfrac{x\left(a^2-x^2\right)^{3/2}}{4} + \dfrac{3a^2x\sqrt{a^2-x^2}}{8} + \dfrac{3}{8} a^4 \arcsin \dfrac {x}{a}+C $ | |
| $ \int x\left(a^2-x^2\right)^{3/2} dx = -\dfrac{x\left(a^2-x^2\right)^{5/2}}{5}+C $ | |
| $ \int x^2\left(a^2-x^2\right)^{3/2} dx = -\dfrac{x\left(a^2-x^2\right)^{5/2}}{6} + \dfrac{a^2x\left(a^2-x^2\right)^{3/2}}{24} + \dfrac{a^4x\sqrt{a^2-x^2}}{16} + \dfrac{a^6}{16} \arcsin \dfrac {x}{a}+C $ | |
| $ \int x^3\left(a^2-x^2\right)^{3/2} dx = \dfrac{\left(a^2-x^2\right)^{7/2}}{7} + \dfrac{a^2\left(a^2-x^2\right)^{5/2}}{5} +C $ | |
| $ \int \dfrac{\left(a^2-x^2\right)^{3/2}}{x} dx = \dfrac{\left(a^2-x^2\right)^{3/2}}{3} + a^2\sqrt{a^2-x^2} - a^3 \ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right)+C $ | |
| $ \int \dfrac{\left(a^2-x^2\right)^{3/2}}{x^2} dx = -\dfrac{\left(a^2-x^2\right)^{3/2}}{x} - \dfrac{3x\sqrt{a^2-x^2}}{2} - \dfrac{3}{2}a^2 \arcsin \dfrac{x}{a}+C $ | |
| $ \int \dfrac{\left(a^2-x^2\right)^{3/2}}{x^3} dx = -\dfrac{\left(a^2-x^2\right)^{3/2}}{2x^2} + \dfrac{3\sqrt{a^2-x^2}}{2} + \dfrac{3}{2}a\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right)+C $ | |
