Danh sách tích phân với hàm lượng giác ngược – Wikipedia tiếng Việt

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

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

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

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

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

${displaystyle left.+{frac {x^{n}{sqrt {1-x^{2}}}-nx^{n-1}sin ^{-1}x}{n-1}}+nint x^{n-2}sin ^{-1}x,dxright)}$

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

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

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

mi>xc−c2ln⁡(c2+x2){displaystyle int arctan {frac {x}{c}},dx=xarctan {frac {x}{c}}-{frac {c}{2}}ln(c^{2}+x^{2})}

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

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

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

${displaystyle int operatorname {arcsec} {frac {x}{c}},dx=xoperatorname {arcsec} {frac {x}{c}}+{frac {x}{c|x|}}ln {|xpm {sqrt {x^{2}-1}}|}}$

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

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

${displaystyle left.left.+(1-n)left(x^{n-1}operatorname {arcsec} {x}+(1-n)int x^{n-2}operatorname {arcsec} {x},dxright)right)right)}$

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

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

${\displaystyle \int <!--\; \int \; -->x^2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}dx=\frac{x^3}{3}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\frac{x}{c}+\frac{c{x}^2}{6}-<!--\; -\; -->\frac{c^3}{6}\ln <!--\; \; -->(c^2+x^2)}$
{displaystyle 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})}

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