Danh sách tích phân với hàm lượng giác ngược

Dưới đây là danh sách các tích phân với hàm lượng giác ngược.

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

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

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

$\int x^n \sin^{-1}x\,dx = \frac{1}{n+1}\left(x^{n+1}\sin^{-1}x \right.$

$\left. + \frac{x^n\sqrt{1 - x^2} - nx^{n-1}\sin^{-1}x}{n-1} + n\int x^{n-2}\sin^{-1}x\,dx\right)$

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

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

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

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

$\int x \arctan\frac{x}{c}\,dx = \frac{c^2+x^2}{2}\arctan\frac{x}{c} - \frac{cx}{2}$

$\int x^2 \arctan\frac{x}{c}\,dx = \frac{x^3}{3}\arctan\frac{x}{c} - \frac{cx^2}{6} + \frac{c^3}{6}\ln{c^2+x^2}$

$\int x^n \arctan\frac{x}{c}\,dx = \frac{x^{n+1}}{n+1}\arctan\frac{x}{c} - \frac{c}{n+1}\int\frac{x^{n+1} dx}{c^2+x^2} \qquad\mbox{(}n\neq 1\mbox{)}$

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

$\int x\arcsec{x}\,dx\,=\,\frac{1}{2}\left(x^2\arcsec{x} - \sqrt{x^2 - 1}\right)$

$\int x^n\arcsec{x}\,dx\,=\,\frac{1}{n+1}\left(x^{n+1}\arcsec{x} - \frac{1}{n}\left(x^{n-1}\sqrt{x^2 - 1}\; \right. \right.$

$\left. \left. + (1-n)\left(x^{n-1}\arcsec{x} + (1-n)\int x^{n-2}\arcsec{x}\,dx \right)\right)\right)$

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

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

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

$\int x^n\,\mathrm{arccot}\,\frac{x}{c}\,dx = \frac{x^{n+1}}{n+1}\,\mathrm{arccot}\,\frac{x}{c} + \frac{c}{n+1}\int\frac{x^{n+1} dx}{c^2+x^2} \qquad\mbox{(}n\neq 1\mbox{)}$

Xem thêm

• Danh sách tích phân

Liên kết ngoài

• Tính biểu thức tích phân