Daftar integral dari fungsi irasional: Perbedaan antara revisi
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Baris 56:
: <math>\int\frac{dx}{xr} = -\frac{1}{a}\,\sinh^{-1}\frac{a}{x} = -\frac{1}{a}\ln\left|\frac{a+r}{x}\right|</math>
== Integral melibatkan <math>s = \sqrt{x^2-a^2}</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>
Baris 117:
= -\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>
== Integral melibatkan <math>t = \sqrt{a^2-x^2}</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] -->
Baris 130:
: <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>
== Integral melibatkan <math>R = \sqrt{ax^2+bx+c}</math> ==
Baris 157:
: <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 -->
== Integral melibatkan <math>S = \sqrt{ax+b}</math> ==
: <math>\int \frac{dx}{x\sqrt{ax + b}}\,=\,\frac{-2}{\sqrt{b}}\tanh^{-1}{\sqrt{\frac{ax + b}{b}}} </math>
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