តារាងអាំងតេក្រាលនៃអនុគមន៍ច្រាស់ត្រីកោណមាត្រ

ខាងក្រោមនេះជាតារាងអាំងតេក្រាល (ព្រីមីទីវ) នៃអនុគមន៍ច្រាស់ត្រីកោណមាត្រ។ សំរាប់តារាងពេញលេញ សូមមើលតារាងអាំងតេក្រាល។ ចំនាំ៖ អក្សរកាត់របស់អនុគមន៍ច្រាស់ត្រីកោណមាត្រអាចសរសេរ​តាមរបៀបបីយ៉ាង។ ឧទាហរណ៍ អនុគមន៍អាកស៊ីនុសអាចសរសេរជា sin⁻¹, asin, ឬ arcsin។ នៅក្នុងអត្ថបទនេះ យើងជ្រើសរើសយកការសរសេរ arcsin ដែលគេនិយមសរសេរ។

អាកស៊ីនុស

$\int \arcsin xdx=x\arcsin x+\sqrt{1-x^2}$

$\int \arcsin \frac{x}{c}dx=x\arcsin \frac{x}{c}+\sqrt{c^2-x^2}$

$\int x\arcsin \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arcsin \frac{x}{c}+\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arcsin \frac{x}{c}dx=\frac{x^3}{3}\arcsin \frac{x}{c}+\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

$\int x^n\arcsin xdx=\frac{1}{n+1}(x^{n+1}\arcsin x+\frac{x^n\sqrt{1-x^2}-n{x}^{n-1}\arcsin x}{n-1}+n\int x^{n-2}\arcsin xdx)$

អាកកូស៊ីនុស

$\int \arccos xdx=x\arccos x-\sqrt{1-x^2}$

$\int \arccos \frac{x}{c}dx=x\arccos \frac{x}{c}-\sqrt{c^2-x^2}$

$\int x\arccos \frac{x}{c}dx=(\frac{x^2}{2}-\frac{c^2}{4})\arccos \frac{x}{c}-\frac{x}{4}\sqrt{c^2-x^2}$

$\int x^2\arccos \frac{x}{c}dx=\frac{x^3}{3}\arccos \frac{x}{c}-\frac{x^2+2c^2}{9}\sqrt{c^2-x^2}$

អាកតង់សង់

$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln |1+x^2|$

$\int \arctan \left(\frac{x}{c}\right)dx=x\arctan \left(\frac{x}{c}\right)-\frac{c}{2}\ln (1+\frac{x^2}{c^2})$

$\int x\arctan \left(\frac{x}{c}\right)dx=\frac{(c^2+x^2)\arctan \left(\frac{x}{c}\right)-cx}{2}$

$\int x^2\arctan \left(\frac{x}{c}\right)dx=\frac{x^3}{3}\arctan \left(\frac{x}{c}\right)-\frac{c{x}^2}{6}+\frac{c^3}{6}\ln |c^2+x^2|$

$\int x^n\arctan \left(\frac{x}{c}\right)dx=\frac{x^{n+1}}{n+1}\arctan \left(\frac{x}{c}\right)-\frac{c}{n+1}\int \frac{x^{n+1}}{c^2+x^2}dx,n\ne -1$

អាកកូសេកង់

$\int \mathrm{arccsc}xdx=x\mathrm{arccsc}x+\ln |x+x\sqrt{\frac{x^2-1}{x^2}}|$

$\int \mathrm{arccsc}\frac{x}{c}dx=x\mathrm{arccsc}\frac{x}{c}+c\ln (\frac{x}{c}(\sqrt{1-\frac{c^2}{x^2}}+1))$

$\int x\mathrm{arccsc}\frac{x}{c}dx=\frac{x^2}{2}\mathrm{arccsc}\frac{x}{c}+\frac{cx}{2}\sqrt{1-\frac{c^2}{x^2}}$

អាកសេកង់

$\int \mathrm{arcsec}xdx=x\mathrm{arcsec}x-\ln |x+x\sqrt{\frac{x^2-1}{x^2}}|$

$\int \mathrm{arcsec}\frac{x}{c}dx=x\mathrm{arcsec}\frac{x}{c}+\frac{x}{c|x|}\ln |x\pm \sqrt{x^2-1}|$

$\int x\mathrm{arcsec}xdx=\frac{1}{2}(x^2\mathrm{arcsec}x-\sqrt{x^2-1})$

$\int x^n\mathrm{arcsec}xdx=\frac{1}{n+1}(x^{n+1}\mathrm{arcsec}x-\frac{1}{n}[x^{n-1}\sqrt{x^2-1}+(1-n)(x^{n-1}\mathrm{arcsec}x+(1-n)\int x^{n-2}\mathrm{arcsec}xdx)])$

អាកកូតង់សង់

$\int \mathrm{arccot}xdx=x\mathrm{arccot}x+\frac{1}{2}\ln |1+x^2|$

$\int \mathrm{arccot}\frac{x}{c}dx=x\mathrm{arccot}\frac{x}{c}+\frac{c}{2}\ln (c^2+x^2)$

$\int x\mathrm{arccot}\frac{x}{c}dx=\frac{c^2+x^2}{2}\mathrm{arccot}\frac{x}{c}+\frac{cx}{2}$

$\int x^2\mathrm{arccot}\frac{x}{c}dx=\frac{x^3}{3}\mathrm{arccot}\frac{x}{c}+\frac{c{x}^2}{6}-\frac{c^3}{6}\ln (c^2+x^2)$

$\int x^n\mathrm{arccot}\frac{x}{c}dx=\frac{x^{n+1}}{n+1}\mathrm{arccot}\frac{x}{c}+\frac{c}{n+1}\int \frac{x^{n+1}}{c^2+x^2}dx,n\ne 1$

ទំព័រគំរូ:តារាងអាំងតេក្រាល