Daftar integral dari fungsi irasional: Perbedaan antara revisi

Artikel ini merupakan '''daftar [[integral]] dari [[fungsi irrasional]]. Untuk daftar integral lainnya, lihat [[tabel integral]].

 

 

: <math>\int r \;dx = \frac{1}{2}\left(x r +a^2\,\ln\left(x+r\right)\right)</math><!-- (1.1) [Abramowitz & Stegun p13 3.3.41] + verified by differentiation -->

: <math>\int\frac{dx}{xr} = -\frac{1}{a}\,\sinh^{-1}\frac{a}{x} = -\frac{1}{a}\ln\left|\frac{a+r}{x}\right|</math>

 

Anggap <math>(x^2>a^2)</math>, untuk <math>(x^2<a^2)</math>, perhatikan bagian berikutnya:

: <math>\int xs\;dx = \frac{1}{3}s^3</math>

= -\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]</math>

 

: <math>\int t \;dx = \frac{1}{2}\left(xt+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}</math><!-- (3.1) [Abramowitz & Stegun p13 3.3.45] -->

 

: <math>\int t\;dx = \frac{1}{2}\left(xt-\sgn x\,\cosh^{-1}\left|\frac{x}{a}\right|\right) \qquad\mbox{(untuk }|x|\ge|a|\mbox{)}</math>

 

 

: <math>\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left|2\sqrt{a}R+2ax+b\right| \qquad \mbox{(untuk }a>0\mbox{)}</math><!-- (4.1) [Abramowitz & Stegun p13 3.3.33] + verified by differentiation -->

: <math>\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\sinh^{-1}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)</math><!-- (4.11) [Abramowitz & Stegun p13 implied by 3.3.38 + 3.3.34] + verified by differentiation -->

 

: <math>\int \frac{dx}{x\sqrt{ax + b}}\,=\,\frac{-2}{\sqrt{b}}\tanh^{-1}{\sqrt{\frac{ax + b}{b}}} </math>

 

: <math>\int x^n \sqrt{ax + b}\,dx \; = \; \frac{2}{2n +1}\left(x^{n+1} \sqrt{ax + b} + bx^{n} \sqrt{ax + b} - nb\int x^{n-1}\sqrt{ax + b}\,dx \right) </math>

 

* Milton Abramowitz and Irene A. Stegun, eds., ''[[Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'' 1972, Dover: New York. ''(See [http://www.math.sfu.ca/~cbm/aands/page_13.htm chapter 3].)''