List of integrals of irrational functions

Below is a list of integrals ( primitive functions) of irrational functions . In the list, the additive integration constant is omitted everywhere.

Integrals with a root of a ² + x ²

Everywhere below: $r={\sqrt {a^{2}+x^{2}}}$ ${\ displaystyle r = {\ sqrt {a ^ {2} + x ^ {2}}}}$ $r={\sqrt {a^{2}+x^{2}}}$ .

\int r\;dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left({x+r}\right)\right)

{\ displaystyle \ int r \; dx = {\ frac {1} {2}} \ left (xr + a ^ {2} \, \ ln \ left ({x + r} \ right) \ right)}

\int r\;dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left({x+r}\right)\right)

\int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {1}{8}}3a^{2}xr+{\frac {3}{8}}a^{4}\ln \left({\frac {x+r}{a}}\right)

{\ displaystyle \ int r ^ {3} \; dx = {\ frac {1} {4}} xr ^ {3} + {\ frac {1} {8}} 3a ^ {2} xr + {\ frac { 3} {8}} a ^ {4} \ ln \ left ({\ frac {x + r} {a}} \ right)}

\int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {1}{8}}3a^{2}xr+{\frac {3}{8}}a^{4}\ln \left({\frac {x+r}{a}}\right)

\int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left({\frac {x+r}{a}}\right)

{\ displaystyle \ int r ^ {5} \; dx = {\ frac {1} {6}} xr ^ {5} + {\ frac {5} {24}} a ^ {2} xr ^ {3} + {\ frac {5} {16}} a ^ {4} xr + {\ frac {5} {16}} a ^ {6} \ ln \ left ({\ frac {x + r} {a}} \ right)}

\int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left({\frac {x+r}{a}}\right)

\int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}

{\ displaystyle \ int xr ^ {2n + 1} \; dx = {\ frac {r ^ {2n + 3}} {2n + 3}}}

\int xr^{{2n+1}}\;dx={\frac {r^{{2n+3}}}{2n+3}}

\int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left({\frac {x+r}{a}}\right)

{\ displaystyle \ int x ^ {2} r \; dx = {\ frac {xr ^ {3}} {4}} - {\ frac {a ^ {2} xr} {8}} - {\ frac { a ^ {4}} {8}} \ ln \ left ({\ frac {x + r} {a}} \ right)}

\int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left({\frac {x+r}{a}}\right)

\int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)

{\ displaystyle \ int x ^ {2} r ^ {3} \; dx = {\ frac {xr ^ {5}} {6}} - {\ frac {a ^ {2} xr ^ {3}} { 24}} - {\ frac {a ^ {4} xr} {16}} - {\ frac {a ^ {6}} {16}} \ ln \ left ({\ frac {x + r} {a} } \ right)}

\int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)

\int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}

{\ displaystyle \ int x ^ {3} r \; dx = {\ frac {r ^ {5}} {5}} - {\ frac {a ^ {2} r ^ {3}} {3}}}

\int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}

\int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}

{\ displaystyle \ int x ^ {3} r ^ {3} \; dx = {\ frac {r ^ {7}} {7}} - {\ frac {a ^ {2} r ^ {5}} { five}}}

\int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}

\int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}

{\ displaystyle \ int x ^ {3} r ^ {2n + 1} \; dx = {\ frac {r ^ {2n + 5}} {2n + 5}} - {\ frac {a ^ {3} r ^ {2n + 3}} {2n + 3}}}

\int x^{3}r^{{2n+1}}\;dx={\frac {r^{{2n+5}}}{2n+5}}-{\frac {a^{3}r^{{2n+3}}}{2n+3}}

\int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)

{\ displaystyle \ int x ^ {4} r \; dx = {\ frac {x ^ {3} r ^ {3}} {6}} - {\ frac {a ^ {2} xr ^ {3}} {8}} + {\ frac {a ^ {4} xr} {16}} + {\ frac {a ^ {6}} {16}} \ ln \ left ({\ frac {x + r} {a }} \ right)}

\int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)

\int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left({\frac {x+r}{a}}\right)

{\ displaystyle \ int x ^ {4} r ^ {3} \; dx = {\ frac {x ^ {3} r ^ {5}} {8}} - {\ frac {a ^ {2} xr ^ {5}} {16}} + {\ frac {a ^ {4} xr ^ {3}} {64}} + {\ frac {3a ^ {6} xr} {128}} + {\ frac {3a ^ {8}} {128}} \ ln \ left ({\ frac {x + r} {a}} \ right)}

\int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left({\frac {x+r}{a}}\right)

\int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}

{\ displaystyle \ int x ^ {5} r \; dx = {\ frac {r ^ {7}} {7}} - {\ frac {2a ^ {2} r ^ {5}} {5}} + {\ frac {a ^ {4} r ^ {3}} {3}}}

\int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}

\int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}

{\ displaystyle \ int x ^ {5} r ^ {3} \; dx = {\ frac {r ^ {9}} {9}} - {\ frac {2a ^ {2} r ^ {7}} { 7}} + {\ frac {a ^ {4} r ^ {5}} {5}}}

\int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}

\int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}

{\ displaystyle \ int x ^ {5} r ^ {2n + 1} \; dx = {\ frac {r ^ {2n + 7}} {2n + 7}} - {\ frac {2a ^ {2} r ^ {2n + 5}} {2n + 5}} + {\ frac {a ^ {4} r ^ {2n + 3}} {2n + 3}}}

\int x^{5}r^{{2n+1}}\;dx={\frac {r^{{2n+7}}}{2n+7}}-{\frac {2a^{2}r^{{2n+5}}}{2n+5}}+{\frac {a^{4}r^{{2n+3}}}{2n+3}}

\int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\operatorname {arsh} {\frac {a}{x}}

{\ displaystyle \ int {\ frac {r \; dx} {x}} = ra \ ln \ left | {\ frac {a + r} {x}} \ right | = ra \ operatorname {arsh} {\ frac {a} {x}}}

\int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\operatorname {arsh} {\frac {a}{x}}

\int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|

{\ displaystyle \ int {\ frac {r ^ {3} \; dx} {x}} = {\ frac {r ^ {3}} {3}} + a ^ {2} ra ^ {3} \ ln \ left | {\ frac {a + r} {x}} \ right |}

\int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|

\int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|

{\ displaystyle \ int {\ frac {r ^ {5} \; dx} {x}} = {\ frac {r ^ {5}} {5}} + {\ frac {a ^ {2} r ^ { 3}} {3}} + a ^ {4} ra ^ {5} \ ln \ left | {\ frac {a + r} {x}} \ right |}

\int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|

{\ displaystyle \ int {\ frac {r ^ {7} \; dx} {x}} = {\ frac {r ^ {7}} {7}} + {\ frac {a ^ {2} r ^ { 5}} {5}} + {\ frac {a ^ {4} r ^ {3}} {3}} + a ^ {6} ra ^ {7} \ ln \ left | {\ frac {a + r } {x}} \ right |}

\int {\frac {dx}{r}}=\operatorname {arsh} {\frac {x}{a}}=\ln \left|x+r\right|

{\ displaystyle \ int {\ frac {dx} {r}} = \ operatorname {arsh} {\ frac {x} {a}} = \ ln \ left | x + r \ right |}

\int {\frac {x\,dx}{r}}=r

{\ displaystyle \ int {\ frac {x \, dx} {r}} = r}

\int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\operatorname {arsh} {\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|

{\ displaystyle \ int {\ frac {x ^ {2} \; dx} {r}} = {\ frac {x} {2}} r - {\ frac {a ^ {2}} {2}} \ , \ operatorname {arsh} {\ frac {x} {a}} = {\ frac {x} {2}} r - {\ frac {a ^ {2}} {2}} \ ln \ left | x + r \ right |}

\int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|

{\ displaystyle \ int {\ frac {dx} {xr}} = - {\ frac {1} {a}} \, \ operatorname {arsh} {\ frac {a} {x}} = - {\ frac { 1} {a}} \ ln \ left | {\ frac {a + r} {x}} \ right |}

Integrals with root of x ² - a ²

Everywhere below: $s={\sqrt {x^{2}-a^{2}}}$ ${\ displaystyle s = {\ sqrt {x ^ {2} -a ^ {2}}}}$ .

Accepted $x^{2}>a^{2}$ ${\ displaystyle x ^ {2}> a ^ {2}}$ for $x^{2}<a^{2}$ ${\ displaystyle x ^ {2} <a ^ {2}}$ see the next section.

\int s\;dx={\frac {1}{2}}\left(xs-a^{2}\ln \left(x+s\right)\right)

{\ displaystyle \ int s \; dx = {\ frac {1} {2}} \ left (xs-a ^ {2} \ ln \ left (x + s \ right) \ right)}

\int xs\;dx={\frac {1}{3}}s^{3}

{\ displaystyle \ int xs \; dx = {\ frac {1} {3}} s ^ {3}}

\int {\frac {s\;dx}{x}}=s-a\cos ^{-1}\left|{\frac {a}{x}}\right|

{\ displaystyle \ int {\ frac {s \; dx} {x}} = sa \ cos ^ {- 1} \ left | {\ frac {a} {x}} \ right |}

\int {\frac {dx}{s}}=\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|x+s\right|

{\ displaystyle \ int {\ frac {dx} {s}} = \ int {\ frac {dx} {\ sqrt {x ^ {2} -a ^ {2}}}} = \ ln \ left | x + s \ right |}

notice, that $\ln \left|{\frac {x+s}{a}}\right|=\mathrm {sgn} (x)\operatorname {arch} \left|{\frac {x}{a}}\right|={\frac {1}{2}}\ln \left({\frac {x+s}{x-s}}\right)$ ${\ displaystyle \ ln \ left | {\ frac {x + s} {a}} \ right | = \ mathrm {sgn} (x) \ operatorname {arch} \ left | {\ frac {x} {a}} \ right | = {\ frac {1} {2}} \ ln \ left ({\ frac {x + s} {xs}} \ right)}$ where $\operatorname {arch} \left|{\frac {x}{a}}\right|$ ${\ displaystyle \ operatorname {arch} \ left | {\ frac {x} {a}} \ right |}$ accepts only positive values.

\int {\frac {x\;dx}{s}}=s

{\ displaystyle \ int {\ frac {x \; dx} {s}} = s}

\int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}

{\ displaystyle \ int {\ frac {x \; dx} {s ^ {3}}} = - {\ frac {1} {s}}}

\int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}

{\ displaystyle \ int {\ frac {x \; dx} {s ^ {5}}} = - {\ frac {1} {3s ^ {3}}}}

\int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}

{\ displaystyle \ int {\ frac {x \; dx} {s ^ {7}}} = - {\ frac {1} {5s ^ {5}}}}

\int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}

{\ displaystyle \ int {\ frac {x \; dx} {s ^ {2n + 1}}} = - {\ frac {1} {(2n-1) s ^ {2n-1}}}}

\int {\frac {x^{2m}\;dx}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\;dx}{s^{2n-1}}}

{\ displaystyle \ int {\ frac {x ^ {2m} \; dx} {s ^ {2n + 1}}} = - {\ frac {1} {2n-1}} {\ frac {x ^ {2m -1}} {s ^ {2n-1}}} + {\ frac {2m-1} {2n-1}} \ int {\ frac {x ^ {2m-2} \; dx} {s ^ { 2n-1}}}}

\int {\frac {x^{2}\;dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|

{\ displaystyle \ int {\ frac {x ^ {2} \; dx} {s}} = {\ frac {xs} {2}} + {\ frac {a ^ {2}} {2}} \ ln \ left | {\ frac {x + s} {a}} \ right |}

\int {\frac {x^{2}\;dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|

{\ displaystyle \ int {\ frac {x ^ {2} \; dx} {s ^ {3}}} = - {\ frac {x} {s}} + \ ln \ left | {\ frac {x + s} {a}} \ right |}

\int {\frac {x^{4}\;dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|

{\ displaystyle \ int {\ frac {x ^ {4} \; dx} {s}} = {\ frac {x ^ {3} s} {4}} + {\ frac {3} {8}} a ^ {2} xs + {\ frac {3} {8}} a ^ {4} \ ln \ left | {\ frac {x + s} {a}} \ right |}

\int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|

{\ displaystyle \ int {\ frac {x ^ {4} \; dx} {s ^ {3}}} = {\ frac {xs} {2}} - {\ frac {a ^ {2} x} { s}} + {\ frac {3} {2}} a ^ {2} \ ln \ left | {\ frac {x + s} {a}} \ right |}

\int {\frac {x^{4}\;dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|

{\ displaystyle \ int {\ frac {x ^ {4} \; dx} {s ^ {5}}} = - {\ frac {x} {s}} - {\ frac {1} {3}} { \ frac {x ^ {3}} {s ^ {3}}} + \ ln \ left | {\ frac {x + s} {a}} \ right |}

\int {\frac {x^{2m}\;dx}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}},

{\ displaystyle \ int {\ frac {x ^ {2m} \; dx} {s ^ {2n + 1}}} = (- 1) ^ {nm} {\ frac {1} {a ^ {2 (nm )}}} \ sum _ {i = 0} ^ {nm-1} {\ frac {1} {2 (m + i) +1}} {nm-1 \ choose i} {\ frac {x ^ { 2 (m + i) +1}} {s ^ {2 (m + i) +1}}},}

Where

n>m\geq 0

{\ displaystyle n> m \ geq 0}

\int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}

{\ displaystyle \ int {\ frac {dx} {s ^ {3}}} = - {\ frac {1} {a ^ {2}}} {\ frac {x} {s}}}

\int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]

{\ displaystyle \ int {\ frac {dx} {s ^ {5}}} = {\ frac {1} {a ^ {4}}} \ left [{\ frac {x} {s}} - {\ frac {1} {3}} {\ frac {x ^ {3}} {s ^ {3}}} \ right]}

\int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]

{\ displaystyle \ int {\ frac {dx} {s ^ {7}}} = - {\ frac {1} {a ^ {6}}} \ left [{\ frac {x} {s}} - { \ frac {2} {3}} {\ frac {x ^ {3}} {s ^ {3}}} + {\ frac {1} {5}} {\ frac {x ^ {5}} {s ^ {5}}} \ right]}

\int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]

{\ displaystyle \ int {\ frac {dx} {s ^ {9}}} = {\ frac {1} {a ^ {8}}} \ left [{\ frac {x} {s}} - {\ frac {3} {3}} {\ frac {x ^ {3}} {s ^ {3}}} + {\ frac {3} {5}} {\ frac {x ^ {5}} {s ^ {5}}} - {\ frac {1} {7}} {\ frac {x ^ {7}} {s ^ {7}}} \ right]}

\int {\frac {x^{2}\;dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}

{\ displaystyle \ int {\ frac {x ^ {2} \; dx} {s ^ {5}}} = - {\ frac {1} {a ^ {2}}} {\ frac {x ^ {3 }} {3s ^ {3}}}}

\int {\frac {x^{2}\;dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]

{\ displaystyle \ int {\ frac {x ^ {2} \; dx} {s ^ {7}}} = {\ frac {1} {a ^ {4}}} \ left [{\ frac {1} {3}} {\ frac {x ^ {3}} {s ^ {3}}} - {\ frac {1} {5}} {\ frac {x ^ {5}} {s ^ {5}} } \ right]}

\int {\frac {x^{2}\;dx}{s^{9}}}=-{\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]

{\ displaystyle \ int {\ frac {x ^ {2} \; dx} {s ^ {9}}} = - {\ 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]}

Integrals with a root of a ² - x ²

Everywhere below: $t={\sqrt {a^{2}-x^{2}}}\qquad {\mbox{(}}|x|\leqslant |a|{\mbox{)}}$ ${\ displaystyle t = {\ sqrt {a ^ {2} -x ^ {2}}} \ qquad {\ mbox {(}} | x | \ leqslant | a | {\ mbox {)}}}$

\int t\;dx={\frac {1}{2}}\left(xt+a^{2}\arcsin {\frac {x}{a}}\right)={\frac {1}{2}}\left(xt-a^{2}\arccos {\frac {x}{a}}\right)

{\ displaystyle \ int t \; dx = {\ frac {1} {2}} \ left (xt + a ^ {2} \ arcsin {\ frac {x} {a}} \ right) = {\ frac { 1} {2}} \ left (xt-a ^ {2} \ arccos {\ frac {x} {a}} \ right)}

\int xt\;dx=-{\frac {1}{3}}t^{3}

{\ displaystyle \ int xt \; dx = - {\ frac {1} {3}} t ^ {3}}

\int {\frac {t\;dx}{x}}=t-a\ln \left|{\frac {a+t}{x}}\right|

{\ displaystyle \ int {\ frac {t \; dx} {x}} = ta \ ln \ left | {\ frac {a + t} {x}} \ right |}

\int {\frac {dx}{t}}=\arcsin {\frac {x}{a}}

{\ displaystyle \ int {\ frac {dx} {t}} = \ arcsin {\ frac {x} {a}}}

\int {\frac {x\;dx}{t}}=-t

{\ displaystyle \ int {\ frac {x \; dx} {t}} = - t}

\int {\frac {x^{2}\;dx}{t}}=-{\frac {x}{2}}t+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}

{\ displaystyle \ int {\ frac {x ^ {2} \; dx} {t}} = - {\ frac {x} {2}} t + {\ frac {a ^ {2}} {2}} \ arcsin {\ frac {x} {a}}}

\int t\;dx={\frac {1}{2}}\left(xt-\operatorname {sgn} x\,\operatorname {arch} \left|{\frac {x}{a}}\right|\right)

{\ displaystyle \ int t \; dx = {\ frac {1} {2}} \ left (xt- \ operatorname {sgn} x \, \ operatorname {arch} \ left | {\ frac {x} {a} } \ right | \ right)}

Integrals with a root from a common square trinomial

It is indicated here: $R=ax^{2}+bx+c$ ${\ displaystyle R = ax ^ {2} + bx + c}$

\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {aR}}+2ax+b\right|\qquad {\mbox{( }}a>0{\mbox{)}}

{\ displaystyle \ int {\ frac {dx} {\ sqrt {ax ^ {2} + bx + c}}} = {\ frac {1} {\ sqrt {a}}} \ ln \ left | 2 {\ sqrt {aR}} + 2ax + b \ right | \ qquad {\ mbox {(}} a> 0 {\ mbox {)}}}

\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{( }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}

{\ displaystyle \ int {\ frac {dx} {\ sqrt {ax ^ {2} + bx + c}}} = {\ frac {1} {\ sqrt {a}}} \, \ operatorname {arsh} { \ frac {2ax + b} {\ sqrt {4ac-b ^ {2}}}} \ qquad {\ mbox {(}} a> 0 {\ mbox {,}} 4ac-b ^ {2}> 0 { \ mbox {)}}}

\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{( }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}

{\ displaystyle \ int {\ frac {dx} {\ sqrt {ax ^ {2} + bx + c}}} = {\ frac {1} {\ sqrt {a}}} \ ln | 2ax + b | \ quad {\ mbox {(}} a> 0 {\ mbox {,}} 4ac-b ^ {2} = 0 {\ mbox {)}}}

\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{( }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}

{\ displaystyle \ int {\ frac {dx} {\ sqrt {ax ^ {2} + bx + c}}} = - {\ frac {1} {\ sqrt {-a}}} \ arcsin {\ frac { 2ax + b} {\ sqrt {b ^ {2} -4ac}}} \ qquad {\ mbox {(}} a <0 {\ mbox {,}} 4ac-b ^ {2} <0 {\ mbox { )}}}

\int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}={\frac {4ax+2b}{(4ac-b^{2}){\sqrt {R}}}}

{\ displaystyle \ int {\ frac {dx} {\ sqrt {(ax ^ {2} + bx + c) ^ {3}}}} = {\ frac {4ax + 2b} {(4ac-b ^ {2 }) {\ sqrt {R}}}}}

\int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{5}}}}={\frac {4ax+2b}{3(4ac-b^{2}){\sqrt {R}}}}\left({\frac {1}{R}}+{\frac {8a}{4ac-b^{2}}}\right)

{\ displaystyle \ int {\ frac {dx} {\ sqrt {(ax ^ {2} + bx + c) ^ {5}}}} = {\ frac {4ax + 2b} {3 (4ac-b ^ { 2}) {\ sqrt {R}}}} \ left ({\ frac {1} {R}} + {\ frac {8a} {4ac-b ^ {2}}} \ right)}

\int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}={\frac {4ax+2b}{(2n-1)(4ac-b^{2})R^{(2n-1)/2}}}+{\frac {8a(n-1)}{(2n-1)(4ac-b^{2})}}\int {\frac {dx}{R^{(2n-1)/2}}}

{\ displaystyle \ int {\ frac {dx} {\ sqrt {(ax ^ {2} + bx + c) ^ {2n + 1}}}} = {\ frac {4ax + 2b} {(2n-1) (4ac-b ^ {2}) R ^ {(2n-1) / 2}}} + {\ frac {8a (n-1)} {(2n-1) (4ac-b ^ {2})} } \ int {\ frac {dx} {R ^ {(2n-1) / 2}}}}

\int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {R}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {R}}}

{\ displaystyle \ int {\ frac {x \; dx} {\ sqrt {ax ^ {2} + bx + c}}} = {\ frac {\ sqrt {R}} {a}} - {\ frac { b} {2a}} \ int {\ frac {dx} {\ sqrt {R}}}}

\int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}=-{\frac {2bx+4c}{(4ac-b^{2}){\sqrt {R}}}}

{\ displaystyle \ int {\ frac {x \; dx} {\ sqrt {(ax ^ {2} + bx + c) ^ {3}}}} = - {\ frac {2bx + 4c} {(4ac- b ^ {2}) {\ sqrt {R}}}}}

\int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}=-{\frac {1}{(2n-1)aR^{(2n-1)/2}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{(2n+1)/2}}}

{\ displaystyle \ int {\ frac {x \; dx} {\ sqrt {(ax ^ {2} + bx + c) ^ {2n + 1}}}} = - {\ frac {1} {(2n- 1) aR ^ {(2n-1) / 2}}} - {\ frac {b} {2a}} \ int {\ frac {dx} {R ^ {(2n + 1) / 2}}}}

\int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {cR}}+bx+2c}{x}}\right)

{\ displaystyle \ int {\ frac {dx} {x {\ sqrt {ax ^ {2} + bx + c}}}} = - {\ frac {1} {\ sqrt {c}}} \ ln \ left ({\ frac {2 {\ sqrt {cR}} + bx + 2c} {x}} \ right)}

\int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)

{\ displaystyle \ int {\ frac {dx} {x {\ sqrt {ax ^ {2} + bx + c}}}} = - {\ frac {1} {\ sqrt {c}}} \ operatorname {arsh } \ left ({\ frac {bx + 2c} {| x | {\ sqrt {4ac-b ^ {2}}}}} \ right)}

Integrals with a root from a linear function

\int {\frac {dx}{x{\sqrt {ax+b}}}}\,=\,{\frac {-2}{\sqrt {b}}}\operatorname {arth} {\sqrt {\frac {ax+b}{b}}}

{\ displaystyle \ int {\ frac {dx} {x {\ sqrt {ax + b}}}}, = \, {\ frac {-2} {\ sqrt {b}}} \ operatorname {arth} { \ sqrt {\ frac {ax + b} {b}}}}

\int {\frac {\sqrt {ax+b}}{x}}\,dx\;=\;2\left({\sqrt {ax+b}}-{\sqrt {b}}\operatorname {arth} {\sqrt {\frac {ax+b}{b}}}\right)

{\ displaystyle \ int {\ frac {\ sqrt {ax + b}} {x}} \, dx \; = \; 2 \ left ({\ sqrt {ax + b}} - {\ sqrt {b}} \ operatorname {arth} {\ sqrt {\ frac {ax + b} {b}}} \ right)}

\int {\frac {x^{n}}{\sqrt {ax+b}}}\,dx\;=\;{\frac {2}{a(2n+1)}}\left(x^{n}{\sqrt {ax+b}}-bn\int {\frac {x^{n-1}}{\sqrt {ax+b}}}\,dx\right)

{\ displaystyle \ int {\ frac {x ^ {n}} {\ sqrt {ax + b}}} \, dx \; = \; {\ frac {2} {a (2n + 1)}} left (x ^ {n} {\ sqrt {ax + b}} - bn \ int {\ frac {x ^ {n-1}} {\ sqrt {ax + b}}}, dx \ right)}

\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)

{\ displaystyle \ 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)}

Bibliography

Books

Gradshtein I. S. Ryzhik I. M. Tables of integrals, sums, series and products. - 4th ed. - M .: Nauka, 1963. - ISBN 0-12-294757-6 // EqWorld
Dvayt G. B. Tables of Integrals of St. Petersburg: Publishing House and Printing House of VNIIG im. B.V. Vedeneeva, 1995 .-- 176 p. - ISBN 5-85529-029-8 .
D. Zwillinger. CRC Standard Mathematical Tables and Formulae , 31st ed., 2002. ISBN 1-58488-291-3 .
M. Abramowitz and IA Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964. ISBN 0-486-61272-4
Korn G.A., Korn T.M. Math reference book for scientists and engineers . - M .: " Science ", 1974.

Integral Tables

Calculation of Integrals

The Integrator (at Wolfram Research )
Empire of Numbers

List of integrals of irrational functions

Integrals with a root of a 2 + x 2

Integrals with root of x 2 - a 2