فرمول های انتگرال (Integral) شامل ریشه ها (Root)، در ریاضیات (Mathematics)
\[ \int \sqrt{x-a}\ dx = \frac{2}{3}(x-a)^{3/2} \] \[ \int \frac{1}{\sqrt{x\pm a}}\ dx = 2\sqrt{x\pm a} \] \[ \int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x} \] \[ \int x\sqrt{x-a}\ dx = \left\{ \begin{array}{l} \frac{2 a}{3} \left({x-a}\right)^{3/2} +\frac{2 }{5}\left( {x-a}\right)^{5/2},\text{ or} \\ \frac{2}{3} x(x-a)^{3/2} - \frac{4}{15} (x-a)^{5/2}, \text{ or} \\ \frac{2}{15}(2a+3x)(x-a)^{3/2} \end{array} \right. \] \[ \int \sqrt{ax+b}\ dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b} \] \[ \int (ax+b)^{3/2}\ dx =\frac{2}{5a}(ax+b)^{5/2} \] \[ \int \frac{x}{\sqrt{x\pm a} } \ dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a} \] \[ \int \sqrt{\frac{x}{a-x}}\ dx = -\sqrt{x(a-x)} -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a} \] \[ \int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right] \] \[ \int x \sqrt{ax + b}\ dx = \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) \sqrt{ax+b} \] \[ \int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} -b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ] \] \[ \int \sqrt{x^3(ax+b)} \ dx =\left [ \frac{b}{12a}- \frac{b^2}{8a^2x}+ \frac{x}{3}\right] \sqrt{x^3(ax+b)} + \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right | \] \[ \int\sqrt{x^2 \pm a^2}\ dx = \frac{1}{2}x\sqrt{x^2\pm a^2} \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | \] \[ \int \sqrt{a^2 - x^2}\ dx = \frac{1}{2} x \sqrt{a^2-x^2} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}} \] \[ \int x \sqrt{x^2 \pm a^2}\ dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2} \] \[ \int \frac{1}{\sqrt{x^2 \pm a^2}}\ dx = \ln \left | x + \sqrt{x^2 \pm a^2} \right | \] \[ \int \frac{1}{\sqrt{a^2 - x^2}}\ dx = \sin^{-1}\frac{x}{a} \] \[ \int \frac{x}{\sqrt{x^2\pm a^2}}\ dx = \sqrt{x^2 \pm a^2} \] \[ \int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2} \] \[ \int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} \mp \frac{1}{2}a^2 \ln \left| x + \sqrt{x^2\pm a^2} \right | \] \[ \int \sqrt{a x^2 + b x + c}\ dx = \frac{b+2ax}{4a}\sqrt{ax^2+bx+c} + \frac{4ac-b^2}{8a^{3/2}}\ln \left| 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right | \] \[ \eqalign{ & \int {x\sqrt {a{x^2} + bx + c}\ dx} = {1 \over {48{a^{5/2}}}}\left( {2\sqrt a \sqrt {a{x^2} + bx + c} \left( { - 3{b^2} + 2abx + 8a\left( {c + a{x^2}} \right)} \right)} \right. \cr & \left. { + 3\left( {{b^3} - 4abc} \right)\ln \left| {b + 2ax + 2\sqrt a \sqrt {a{x^2} + bx + c} } \right|} \right) \cr} \] \[ \int\frac{1}{\sqrt{ax^2+bx+c}}\ dx= \frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \] \[ \int \frac{x}{\sqrt{ax^2+bx+c}}\ dx= \frac{1}{a}\sqrt{ax^2+bx + c} - \frac{b}{2a^{3/2}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \] \[ \int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}} \]
نویسنده علیرضا گلمکانی
شماره کلید 10086
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