Tabelul integralelor nedeterminate ale funcțiilor raționale

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Această pagină conține un tabel de integrale nedeterminate ale funcțiilor raționale . $C.$ ${\ displaystyle C}$ $C.$ denotă o constantă arbitrară de integrare care are sens să se specifice doar în raport cu o specificație a valorii integralei la un moment dat.

Pentru alte integrale a se vedea Integral § Tabelele de integrale .

\int (ax+b)^{n}\mathrm {d} x={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\mbox{(per }}n\neq -1{\mbox{)}}

{\ displaystyle \ int (ax + b) ^ {n} \ mathrm {d} x = {\ frac {(ax + b) ^ {n + 1}} {a (n + 1)}} + C \ qquad {\ mbox {(for}} n \ neq -1 {\ mbox {)}}}

{\ displaystyle \ int (ax + b) ^ {n} \ mathrm {d} x = {\ frac {(ax + b) ^ {n + 1}} {a (n + 1)}} + C \ qquad {\ mbox {(for}} n \ neq -1 {\ mbox {)}}}

\int x^{n-1}(ax^{n}+b)^{c}\;\mathrm {d} x={\frac {(ax^{n}+b)^{c+1}}{na(c+1)}}+C\qquad {\mbox{(per }}n\neq 0{\mbox{)}}

{\ displaystyle \ int x ^ {n-1} (ax ^ {n} + b) ^ {c} \; \ mathrm {d} x = {\ frac {(ax ^ {n} + b) ^ {c +1}} {na (c + 1)}} + C \ qquad {\ mbox {(pentru}} n \ neq 0 {\ mbox {)}}}

{\ displaystyle \ int x ^ {n-1} (ax ^ {n} + b) ^ {c} \; \ mathrm {d} x = {\ frac {(ax ^ {n} + b) ^ {c +1}} {na (c + 1)}} + C \ qquad {\ mbox {(pentru}} n \ neq 0 {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{ax+b}}={\frac {1}{a}}\ln \left|ax+b\right|+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax + b}} = {\ frac {1} {a}} \ ln \ left | ax + b \ right | + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax + b}} = {\ frac {1} {a}} \ ln \ left | ax + b \ right | + C}

\int x(ax+b)^{n}\mathrm {d} x={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}+C\qquad {\mbox{(per }}n\not \in \{-1,-2\}{\mbox{)}}

{\ displaystyle \ int x (ax + b) ^ {n} \ mathrm {d} x = {\ frac {a (n + 1) xb} {a ^ {2} (n + 1) (n + 2) }} (ax + b) ^ {n + 1} + C \ qquad {\ mbox {(pentru}} n \ not \ in \ {- 1, -2 \} {\ mbox {)}}}

{\ displaystyle \ int x (ax + b) ^ {n} \ mathrm {d} x = {\ frac {a (n + 1) xb} {a ^ {2} (n + 1) (n + 2) }} (ax + b) ^ {n + 1} + C \ qquad {\ mbox {(pentru}} n \ not \ in \ {- 1, -2 \} {\ mbox {)}}}

\int {\frac {x\;\mathrm {d} x}{ax+b}}={\frac {x}{a}}-{\frac {b}{a^{2}}}\log \left|ax+b\right|+C

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {ax + b}} = {\ frac {x} {a}} - {\ frac {b} {a ^ {2}} } \ log \ left | ax + b \ right | + C}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {ax + b}} = {\ frac {x} {a}} - {\ frac {b} {a ^ {2}} } \ log \ left | ax + b \ right | + C}

\int {\frac {x\;\mathrm {d} x}{(ax+b)^{2}}}={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\log \left|ax+b\right|+C

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {(ax + b) ^ {2}}} = {\ frac {b} {a ^ {2} (ax + b)} } + {\ frac {1} {a ^ {2}}} \ log \ left | ax + b \ right | + C}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {(ax + b) ^ {2}}} = {\ frac {b} {a ^ {2} (ax + b)} } + {\ frac {1} {a ^ {2}}} \ log \ left | ax + b \ right | + C}

\int {\frac {x\;\mathrm {d} x}{(ax+b)^{n}}}={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}+C\qquad {\mbox{(per }}n\not \in \{-1,-2\}{\mbox{)}}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {(ax + b) ^ {n}}} = {\ frac {a (1-n) xb} {a ^ {2} (n-1) (n-2) (ax + b) ^ {n-1}}} + C \ qquad {\ mbox {(pentru}} n \ not \ in \ {- 1, -2 \} { \ mbox {)}}}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {(ax + b) ^ {n}}} = {\ frac {a (1-n) xb} {a ^ {2} (n-1) (n-2) (ax + b) ^ {n-1}}} + C \ qquad {\ mbox {(pentru}} n \ not \ in \ {- 1, -2 \} { \ mbox {)}}}

\int {\frac {x^{2}\;\mathrm {d} x}{ax+b}}={\frac {1}{a^{3}}}\left[{\frac {(ax+b)^{2}}{2}}-2b(ax+b)+b^{2}\log \left|ax+b\right|\right]+C

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {ax + b}} = {\ frac {1} {a ^ {3}}} \ left [{\ frac {(ax + b) ^ {2}} {2}} - 2b (ax + b) + b ^ {2} \ log \ left | ax + b \ right | \ right] + C}

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {ax + b}} = {\ frac {1} {a ^ {3}}} \ left [{\ frac {(ax + b) ^ {2}} {2}} - 2b (ax + b) + b ^ {2} \ log \ left | ax + b \ right | \ right] + C}

\int {\frac {x^{2}\;\mathrm {d} x}{(ax+b)^{2}}}={\frac {1}{a^{3}}}\left(ax+b-2b\log \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)+C

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {(ax + b) ^ {2}}} = {\ frac {1} {a ^ {3}}} \ left (ax + b-2b \ log \ left | ax + b \ right | - {\ frac {b ^ {2}} {ax + b}} \ right) + C}

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {(ax + b) ^ {2}}} = {\ frac {1} {a ^ {3}}} \ left (ax + b-2b \ log \ left | ax + b \ right | - {\ frac {b ^ {2}} {ax + b}} \ right) + C}

\int {\frac {x^{2}\;\mathrm {d} x}{(ax+b)^{3}}}={\frac {1}{a^{3}}}\left[\log \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right]+C

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {(ax + b) ^ {3}}} = {\ frac {1} {a ^ {3}}} \ left [\ log \ left | ax + b \ right | + {\ frac {2b} {ax + b}} - {\ frac {b ^ {2}} {2 (ax + b) ^ {2}} } \ dreapta] + C}

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {(ax + b) ^ {3}}} = {\ frac {1} {a ^ {3}}} \ left [\ log \ left | ax + b \ right | + {\ frac {2b} {ax + b}} - {\ frac {b ^ {2}} {2 (ax + b) ^ {2}} } \ dreapta] + C}

\int {\frac {x^{2}\;\mathrm {d} x}{(ax+b)^{n}}}={\frac {1}{a^{3}}}\left[-{\frac {1}{(n-3)(ax+b)^{n-3}}}+{\frac {2b}{(n-2)(a+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right]+C\qquad {\mbox{(per }}n\not \in \{1,2,3\}{\mbox{)}}

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {(ax + b) ^ {n}}} = {\ frac {1} {a ^ {3}}} \ left [- {\ frac {1} {(n-3) (ax + b) ^ {n-3}}} + {\ frac {2b} {(n-2) (a + b) ^ {n -2}}} - {\ frac {b ^ {2}} {(n-1) (ax + b) ^ {n-1}}} \ right] + C \ qquad {\ mbox {(for}} n \ not \ in \ {1,2,3 \} {\ mbox {)}}}

{\ displaystyle \ int {\ frac {x ^ {2} \; \ mathrm {d} x} {(ax + b) ^ {n}}} = {\ frac {1} {a ^ {3}}} \ left [- {\ frac {1} {(n-3) (ax + b) ^ {n-3}}} + {\ frac {2b} {(n-2) (a + b) ^ {n -2}}} - {\ frac {b ^ {2}} {(n-1) (ax + b) ^ {n-1}}} \ right] + C \ qquad {\ mbox {(for}} n \ not \ in \ {1,2,3 \} {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{x(ax+b)}}=-{\frac {1}{b}}\log \left|{\frac {ax+b}{x}}\right|+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x (ax + b)}} = - {\ frac {1} {b}} \ log \ left | {\ frac {ax + b} {x}} \ right | + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x (ax + b)}} = - {\ frac {1} {b}} \ log \ left | {\ frac {ax + b} {x}} \ right | + C}

\int {\frac {\mathrm {d} x}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\log \left|{\frac {ax+b}{x}}\right|+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} (ax + b)}} = - {\ frac {1} {bx}} + {\ frac {a} {b ^ {2}}} \ log \ left | {\ frac {ax + b} {x}} \ right | + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} (ax + b)}} = - {\ frac {1} {bx}} + {\ frac {a} {b ^ {2}}} \ log \ left | {\ frac {ax + b} {x}} \ right | + C}

\int {\frac {\mathrm {d} x}{x^{2}(ax+b)^{2}}}=-a\left[{\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2}x}}-{\frac {2}{b^{3}}}\log \left|{\frac {ax+b}{x}}\right|\right]+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} (ax + b) ^ {2}}} = - a \ left [{\ frac {1} {b ^ {2 } (ax + b)}} + {\ frac {1} {ab ^ {2} x}} - {\ frac {2} {b ^ {3}}} \ log \ left | {\ frac {ax + b} {x}} \ right | \ right] + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} (ax + b) ^ {2}}} = - a \ left [{\ frac {1} {b ^ {2 } (ax + b)}} + {\ frac {1} {ab ^ {2} x}} - {\ frac {2} {b ^ {3}}} \ log \ left | {\ frac {ax + b} {x}} \ right | \ right] + C}

\int {\frac {\mathrm {d} x}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} + a ^ {2}}} = {\ frac {1} {a}} \ arctan {\ frac {x} { a}} + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} + a ^ {2}}} = {\ frac {1} {a}} \ arctan {\ frac {x} { a}} + C}

\int {\frac {\mathrm {d} x}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {settanh} {\frac {x}{a}}={\frac {1}{2a}}\log {\frac {a-x}{a+x}}+C\qquad {\mbox{(per }}|x|<|a|{\mbox{)}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} -a ^ {2}}} = - {\ frac {1} {a}} \, \ mathrm {settanh} { \ frac {x} {a}} = {\ frac {1} {2a}} \ log {\ frac {ax} {a + x}} + C \ qquad {\ mbox {(for}} | x | < | a | {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} -a ^ {2}}} = - {\ frac {1} {a}} \, \ mathrm {settanh} { \ frac {x} {a}} = {\ frac {1} {2a}} \ log {\ frac {ax} {a + x}} + C \ qquad {\ mbox {(for}} | x | < | a | {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {settcoth} {\frac {x}{a}}={\frac {1}{2a}}\log {\frac {x-a}{x+a}}+C\qquad {\mbox{(per }}|x|>|a|{\mbox{)}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} -a ^ {2}}} = - {\ frac {1} {a}} \, \ mathrm {settcoth} { \ frac {x} {a}} = {\ frac {1} {2a}} \ log {\ frac {xa} {x + a}} + C \ qquad {\ mbox {(for}} | x |> | a | {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2} -a ^ {2}}} = - {\ frac {1} {a}} \, \ mathrm {settcoth} { \ frac {x} {a}} = {\ frac {1} {2a}} \ log {\ frac {xa} {x + a}} + C \ qquad {\ mbox {(for}} | x |> | a | {\ mbox {)}}}

În următoarele formule se dorește a fi $a\neq 0$ ${\ displaystyle a \ neq 0}$ $a \ neq 0$

\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+C\qquad {\mbox{(per }}4ac-b^{2}>0{\mbox{)}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}} = {\ frac {2} {\ sqrt {4ac-b ^ {2}}}} \ arctan {\ frac {2ax + b} {\ sqrt {4ac-b ^ {2}}}} + C \ qquad {\ mbox {(per}} 4ac-b ^ {2}> 0 {\ mbox {)} }}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}} = {\ frac {2} {\ sqrt {4ac-b ^ {2}}}} \ arctan {\ frac {2ax + b} {\ sqrt {4ac-b ^ {2}}}} + C \ qquad {\ mbox {(per}} 4ac-b ^ {2}> 0 {\ mbox {)} }}

\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}+C\qquad {\mbox{(per }}4ac-b^{2}=0{\mbox{)}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}} = - {\ frac {2} {2ax + b}} + C \ qquad {\ mbox { (pentru}} 4ac-b ^ {2} = 0 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}} = - {\ frac {2} {2ax + b}} + C \ qquad {\ mbox { (pentru}} 4ac-b ^ {2} = 0 {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}=-{\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {settanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\log \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|+C\qquad {\mbox{(per }}4ac-b^{2}<0{\mbox{)}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}} = - {\ frac {2} {\ sqrt {b ^ {2} -4ac}}} \, \ mathrm {settanh} {\ frac {2ax + b} {\ sqrt {b ^ {2} -4ac}}} = {\ frac {1} {\ sqrt {b ^ {2} -4ac}}} \ log \ left | {\ frac {2ax + b - {\ sqrt {b ^ {2} -4ac}}} {2ax + b + {\ sqrt {b ^ {2} -4ac}}}} \ right | + C \ qquad {\ mbox {(per}} 4ac-b ^ {2} <0 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}} = - {\ frac {2} {\ sqrt {b ^ {2} -4ac}}} \, \ mathrm {settanh} {\ frac {2ax + b} {\ sqrt {b ^ {2} -4ac}}} = {\ frac {1} {\ sqrt {b ^ {2} -4ac}}} \ log \ left | {\ frac {2ax + b - {\ sqrt {b ^ {2} -4ac}}} {2ax + b + {\ sqrt {b ^ {2} -4ac}}}} \ right | + C \ qquad {\ mbox {(per}} 4ac-b ^ {2} <0 {\ mbox {)}}}

\int {\frac {x\;\mathrm {d} x}{ax^{2}+bx+c}}={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {ax ^ {2} + bx + c}} = {\ frac {1} {2a}} \ ln \ left | ax ^ { 2} + bx + c \ right | - {\ frac {b} {2a}} \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}}}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {ax ^ {2} + bx + c}} = {\ frac {1} {2a}} \ ln \ left | ax ^ { 2} + bx + c \ right | - {\ frac {b} {2a}} \ int {\ frac {\ mathrm {d} x} {ax ^ {2} + bx + c}}}

\int {\frac {mx+n}{ax^{2}+bx+c}}\mathrm {d} x={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {4ac-b^{2}}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+C\qquad {\mbox{(per }}4ac-b^{2}>0{\mbox{)}}

{\ displaystyle \ int {\ frac {mx + n} {ax ^ {2} + bx + c}} \ mathrm {d} x = {\ frac {m} {2a}} \ ln \ left | ax ^ { 2} + bx + c \ right | + {\ frac {2an-bm} {a {\ sqrt {4ac-b ^ {2}}}}} \ arctan {\ frac {2ax + b} {\ sqrt {4ac -b ^ {2}}}} + C \ qquad {\ mbox {(pentru}} 4ac-b ^ {2}> 0 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {mx + n} {ax ^ {2} + bx + c}} \ mathrm {d} x = {\ frac {m} {2a}} \ ln \ left | ax ^ { 2} + bx + c \ right | + {\ frac {2an-bm} {a {\ sqrt {4ac-b ^ {2}}}}} \ arctan {\ frac {2ax + b} {\ sqrt {4ac -b ^ {2}}}} + C \ qquad {\ mbox {(pentru}} 4ac-b ^ {2}> 0 {\ mbox {)}}}

\int {\frac {mx+n}{ax^{2}+bx+c}}\mathrm {d} x={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {setttanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+C\qquad {\mbox{(per }}4ac-b^{2}<0{\mbox{)}}

{\ displaystyle \ int {\ frac {mx + n} {ax ^ {2} + bx + c}} \ mathrm {d} x = {\ frac {m} {2a}} \ ln \ left | ax ^ { 2} + bx + c \ right | + {\ frac {2an-bm} {a {\ sqrt {b ^ {2} -4ac}}}} \, \ mathrm {setttanh} {\ frac {2ax + b} {\ sqrt {b ^ {2} -4ac}}} + C \ qquad {\ mbox {(per}} 4ac-b ^ {2} <0 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {mx + n} {ax ^ {2} + bx + c}} \ mathrm {d} x = {\ frac {m} {2a}} \ ln \ left | ax ^ { 2} + bx + c \ right | + {\ frac {2an-bm} {a {\ sqrt {b ^ {2} -4ac}}}} \, \ mathrm {setttanh} {\ frac {2ax + b} {\ sqrt {b ^ {2} -4ac}}} + C \ qquad {\ mbox {(per}} 4ac-b ^ {2} <0 {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{(ax^{2}+bx+c)^{n}}}={\frac {2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}+{\frac {(2n-3)2a}{(n-1)(4ac-b^{2})}}\int {\frac {\mathrm {d} x}{(ax^{2}+bx+c)^{n-1}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n}}} = {\ frac {2ax + b} {(n-1) ( 4ac-b ^ {2}) (ax ^ {2} + bx + c) ^ {n-1}}} + {\ frac {(2n-3) 2a} {(n-1) (4ac-b ^ {2})}} \ int {\ frac {\ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n-1}}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n}}} = {\ frac {2ax + b} {(n-1) ( 4ac-b ^ {2}) (ax ^ {2} + bx + c) ^ {n-1}}} + {\ frac {(2n-3) 2a} {(n-1) (4ac-b ^ {2})}} \ int {\ frac {\ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n-1}}}}

\int {\frac {x\;\mathrm {d} x}{(ax^{2}+bx+c)^{n}}}=-{\frac {bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}-{\frac {b(2n-3)}{(n-1)(4ac-b^{2})}}\int {\frac {\mathrm {d} x}{(ax^{2}+bx+c)^{n-1}}}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n}}} = - {\ frac {bx + 2c} {(n -1) (4ac-b ^ {2}) (ax ^ {2} + bx + c) ^ {n-1}}} - {\ frac {b (2n-3)} {(n-1) ( 4ac-b ^ {2})}} \ int {\ frac {\ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n-1}}}}

{\ displaystyle \ int {\ frac {x \; \ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n}}} = - {\ frac {bx + 2c} {(n -1) (4ac-b ^ {2}) (ax ^ {2} + bx + c) ^ {n-1}}} - {\ frac {b (2n-3)} {(n-1) ( 4ac-b ^ {2})}} \ int {\ frac {\ mathrm {d} x} {(ax ^ {2} + bx + c) ^ {n-1}}}}

\int {\frac {\mathrm {d} x}{x(ax^{2}+bx+c)}}={\frac {1}{2c}}\log \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x (ax ^ {2} + bx + c)}} = {\ frac {1} {2c}} \ log \ left | {\ frac {x ^ {2}} {ax ^ {2} + bx + c}} \ right | - {\ frac {b} {2c}} \ int {\ frac {\ mathrm {d} x} {ax ^ { 2} + bx + c}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x (ax ^ {2} + bx + c)}} = {\ frac {1} {2c}} \ log \ left | {\ frac {x ^ {2}} {ax ^ {2} + bx + c}} \ right | - {\ frac {b} {2c}} \ int {\ frac {\ mathrm {d} x} {ax ^ { 2} + bx + c}}}

\int {\frac {\mathrm {d} x}{x^{4}+1}}={\frac {1}{2{\sqrt {2}}}}\left[\arctan({\sqrt {2}}x+1)+\arctan({\sqrt {2}}x-1)\right]+{\frac {1}{4{\sqrt {2}}}}\left[\log |x^{2}+{\sqrt {2}}x+1|-\log |x^{2}-{\sqrt {2}}x+1|\right]+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {4} +1}} = {\ frac {1} {2 {\ sqrt {2}}}} \ left [\ arctan ( {\ sqrt {2}} x + 1) + \ arctan ({\ sqrt {2}} x-1) \ right] + {\ frac {1} {4 {\ sqrt {2}}}} \ left [ \ log | x ^ {2} + {\ sqrt {2}} x + 1 | - \ log | x ^ {2} - {\ sqrt {2}} x + 1 | \ right] + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {4} +1}} = {\ frac {1} {2 {\ sqrt {2}}}} \ left [\ arctan ( {\ sqrt {2}} x + 1) + \ arctan ({\ sqrt {2}} x-1) \ right] + {\ frac {1} {4 {\ sqrt {2}}}} \ left [ \ log | x ^ {2} + {\ sqrt {2}} x + 1 | - \ log | x ^ {2} - {\ sqrt {2}} x + 1 | \ right] + C}

\int {\frac {\mathrm {d} x}{x^{2^{n}}+1}}=\sum _{k=1}^{2^{n-1}}\left\{{\frac {1}{2^{n-1}}}\sin {\frac {(2k-1)\pi }{2^{n}}}\cdot \arctan \left[\left(x-\cos {\frac {(2k-1)\pi }{2^{n}}}\right)\csc {\frac {(2k-1)\pi }{2^{n}}}\right]-{\frac {1}{2^{n}}}\cos {\frac {(2k-1)\pi }{2^{n}}}\cdot \log \left|x^{2}-2x\cos {\frac {(2k-1)\pi }{2^{n}}}+1\right|\right\}+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2 ^ {n}} + 1}} = \ sum _ {k = 1} ^ {2 ^ {n-1}} \ left \ {{\ frac {1} {2 ^ {n-1}}} \ sin {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ cdot \ arctan \ left [\ left (x- \ cos {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ right) \ csc {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ dreapta] - {\ frac {1} {2 ^ {n}}} \ cos {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ cdot \ log \ left | x ^ { 2} -2x \ cos {\ frac {(2k-1) \ pi} {2 ^ {n}}} + 1 \ right | \ right \} + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {2 ^ {n}} + 1}} = \ sum _ {k = 1} ^ {2 ^ {n-1}} \ left \ {{\ frac {1} {2 ^ {n-1}}} \ sin {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ cdot \ arctan \ left [\ left (x- \ cos {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ right) \ csc {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ dreapta] - {\ frac {1} {2 ^ {n}}} \ cos {\ frac {(2k-1) \ pi} {2 ^ {n}}} \ cdot \ log \ left | x ^ { 2} -2x \ cos {\ frac {(2k-1) \ pi} {2 ^ {n}}} + 1 \ right | \ right \} + C}

Din fiecare funcție rațională este posibil să se găsească integralul nedefinit prin descompunerea ei într-o sumă de funcții ale formei

{\frac {ex+f}{\left(ax^{2}+bx+c\right)^{n}}}

{\ displaystyle {\ frac {ex + f} {\ left (ax ^ {2} + bx + c \ right) ^ {n}}}}

{\ displaystyle {\ frac {ex + f} {\ left (ax ^ {2} + bx + c \ right) ^ {n}}}}

și aplicarea unor formule anterioare diferitelor adaosuri.

Bibliografie

Murray R. Spiegel, Manual de matematică , Etas Libri, 1974, pp. 60-74.

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Tabelele integrale