Tabel cu integrale nedeterminate ale funcțiilor logaritmice

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Această pagină conține un tabel de integrale nedeterminate ale funcțiilor logaritmice . Pentru alte integrale a se vedea Integral § Tabelele de integrale .

În această pagină se presupune că x este o variabilă pe setul de reali pozitivi. C reprezintă o constantă de integrare arbitrară, care poate fi specificată numai pentru o anumită valoare a integralei.

\int \log cx\,\mathrm {d} x=x\log cx-x+C

{\ displaystyle \ int \ log cx \, \ mathrm {d} x = x \ log cx-x + C}

{\ displaystyle \ int \ log cx \, \ mathrm {d} x = x \ log cx-x + C}

\int (\log x)^{2}\;\mathrm {d} x=x(\log x)^{2}-2x\log x+2x+C

{\ displaystyle \ int (\ log x) ^ {2} \; \ mathrm {d} x = x (\ log x) ^ {2} -2x \ log x + 2x + C}

{\ displaystyle \ int (\ log x) ^ {2} \; \ mathrm {d} x = x (\ log x) ^ {2} -2x \ log x + 2x + C}

\int (\log cx)^{n}\;\mathrm {d} x=x(\log cx)^{n}-n\int (\log cx)^{n-1}\,\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int (\ log cx) ^ {n} \; \ mathrm {d} x = x (\ log cx) ^ {n} -n \ int (\ log cx) ^ {n-1} \, \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int (\ log cx) ^ {n} \; \ mathrm {d} x = x (\ log cx) ^ {n} -n \ int (\ log cx) ^ {n-1} \, \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{\log x}}=\log |\log x|+\log x+\sum _{k=2}^{\infty }{\frac {(\log x)^{k}}{k\cdot k!}}+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ log x}} = \ log | \ log x | + \ log x + \ sum _ {k = 2} ^ {\ infty} {\ frac {(\ log x) ^ {k}} {k \ cdot k!}} + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ log x}} = \ log | \ log x | + \ log x + \ sum _ {k = 2} ^ {\ infty} {\ frac {(\ log x) ^ {k}} {k \ cdot k!}} + C}

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

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {(\ log x) ^ {n}}} = - {\ frac {x} {(n-1) (\ ln x) ^ {n -1}}} + {\ frac {1} {n-1}} \ int {\ frac {\ mathrm {d} x} {(\ log x) ^ {n-1}}} + C \ qquad { \ mbox {(pentru}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {(\ log x) ^ {n}}} = - {\ frac {x} {(n-1) (\ ln x) ^ {n -1}}} + {\ frac {1} {n-1}} \ int {\ frac {\ mathrm {d} x} {(\ log x) ^ {n-1}}} + C \ qquad { \ mbox {(pentru}} n \ neq 1 {\ mbox {)}}}

\int x^{m}\log x\;\mathrm {d} x=x^{m+1}\left({\frac {\log x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)+C\qquad {\mbox{(per }}m\neq -1{\mbox{)}}

{\ displaystyle \ int x ^ {m} \ log x \; \ mathrm {d} x = x ^ {m + 1} \ left ({\ frac {\ log x} {m + 1}} - {\ frac {1} {(m + 1) ^ {2}}} \ right) + C \ qquad {\ mbox {(for}} m \ neq -1 {\ mbox {)}}}

{\ displaystyle \ int x ^ {m} \ log x \; \ mathrm {d} x = x ^ {m + 1} \ left ({\ frac {\ log x} {m + 1}} - {\ frac {1} {(m + 1) ^ {2}}} \ right) + C \ qquad {\ mbox {(for}} m \ neq -1 {\ mbox {)}}}

\int x^{m}(\log x)^{n}\;\mathrm {d} x={\frac {x^{m+1}(\log x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\log x)^{n-1}\mathrm {d} x+C\qquad {\mbox{(per }}m,n\neq 1{\mbox{)}}

{\ displaystyle \ int x ^ {m} (\ log x) ^ {n} \; \ mathrm {d} x = {\ frac {x ^ {m + 1} (\ log x) ^ {n}} { m + 1}} - {\ frac {n} {m + 1}} \ int x ^ {m} (\ log x) ^ {n-1} \ mathrm {d} x + C \ qquad {\ mbox { (pentru}} m, n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int x ^ {m} (\ log x) ^ {n} \; \ mathrm {d} x = {\ frac {x ^ {m + 1} (\ log x) ^ {n}} { m + 1}} - {\ frac {n} {m + 1}} \ int x ^ {m} (\ log x) ^ {n-1} \ mathrm {d} x + C \ qquad {\ mbox { (pentru}} m, n \ neq 1 {\ mbox {)}}}

\int {\frac {(\log x)^{n}\;\mathrm {d} x}{x}}={\frac {(\log x)^{n+1}}{n+1}}+C

{\ displaystyle \ int {\ frac {(\ log x) ^ {n} \; \ mathrm {d} x} {x}} = {\ frac {(\ log x) ^ {n + 1}} {n +1}} + C}

{\ displaystyle \ int {\ frac {(\ log x) ^ {n} \; \ mathrm {d} x} {x}} = {\ frac {(\ log x) ^ {n + 1}} {n +1}} + C}

\int {\frac {\log x\,\mathrm {d} x}{x^{m}}}=-{\frac {\log x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}+C\qquad {\mbox{(per }}m\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {\ log x \, \ mathrm {d} x} {x ^ {m}}} = - {\ frac {\ log x} {(m-1) x ^ {m- 1}}} - {\ frac {1} {(m-1) ^ {2} x ^ {m-1}}} + C \ qquad {\ mbox {(pentru}} m \ neq 1 {\ mbox { )}}}

{\ displaystyle \ int {\ frac {\ log x \, \ mathrm {d} x} {x ^ {m}}} = - {\ frac {\ log x} {(m-1) x ^ {m- 1}}} - {\ frac {1} {(m-1) ^ {2} x ^ {m-1}}} + C \ qquad {\ mbox {(pentru}} m \ neq 1 {\ mbox { )}}}

\int {\frac {(\log x)^{n}\;\mathrm {d} x}{x^{m}}}=-{\frac {(\log x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\log x)^{n-1}\mathrm {d} x}{x^{m}}}+C\qquad {\mbox{(per }}m,n\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {(\ log x) ^ {n} \; \ mathrm {d} x} {x ^ {m}}} = - {\ frac {(\ log x) ^ {n} } {(m-1) x ^ {m-1}}} + {\ frac {n} {m-1}} \ int {\ frac {(\ log x) ^ {n-1} \ mathrm {d } x} {x ^ {m}}} + C \ qquad {\ mbox {(pentru}} m, n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {(\ log x) ^ {n} \; \ mathrm {d} x} {x ^ {m}}} = - {\ frac {(\ log x) ^ {n} } {(m-1) x ^ {m-1}}} + {\ frac {n} {m-1}} \ int {\ frac {(\ log x) ^ {n-1} \ mathrm {d } x} {x ^ {m}}} + C \ qquad {\ mbox {(pentru}} m, n \ neq 1 {\ mbox {)}}}

\int {\frac {x^{m}\;\mathrm {d} x}{(\log x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\log x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}\mathrm {d} x}{(\log x)^{n-1}}}+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {x ^ {m} \; \ mathrm {d} x} {(\ log x) ^ {n}}} = - {\ frac {x ^ {m + 1}} { (n-1) (\ log x) ^ {n-1}}} + {\ frac {m + 1} {n-1}} \ int {\ frac {x ^ {m} \ mathrm {d} x } {(\ log x) ^ {n-1}}} + C \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {x ^ {m} \; \ mathrm {d} x} {(\ log x) ^ {n}}} = - {\ frac {x ^ {m + 1}} { (n-1) (\ log x) ^ {n-1}}} + {\ frac {m + 1} {n-1}} \ int {\ frac {x ^ {m} \ mathrm {d} x } {(\ log x) ^ {n-1}}} + C \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{x\log x}}=\log |\log x|+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x \ log x}} = \ log | \ log x | + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x \ log x}} = \ log | \ log x | + C}

\int {\frac {\mathrm {d} x}{x^{n}\log x}}=\log |\log x|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(n-1)^{k}(\log x)^{k}}{k\cdot k!}}+C

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {n} \ log x}} = \ log | \ log x | + \ sum _ {k = 1} ^ {\ infty} ( -1) ^ {k} {\ frac {(n-1) ^ {k} (\ log x) ^ {k}} {k \ cdot k!}} + C}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x ^ {n} \ log x}} = \ log | \ log x | + \ sum _ {k = 1} ^ {\ infty} ( -1) ^ {k} {\ frac {(n-1) ^ {k} (\ log x) ^ {k}} {k \ cdot k!}} + C}

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

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x (\ log x) ^ {n}}} = - {\ frac {1} {(n-1) (\ ln x) ^ { n-1}}} + C \ qquad {\ mbox {(pentru}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {x (\ log x) ^ {n}}} = - {\ frac {1} {(n-1) (\ ln x) ^ { n-1}}} + C \ qquad {\ mbox {(pentru}} n \ neq 1 {\ mbox {)}}}

\int \sin(\log x)\;\mathrm {d} x={\frac {x}{2}}[\sin(\log x)-\cos(\log x)]+C

{\ displaystyle \ int \ sin (\ log x) \; \ mathrm {d} x = {\ frac {x} {2}} [\ sin (\ log x) - \ cos (\ log x)] + C }

{\ displaystyle \ int \ sin (\ log x) \; \ mathrm {d} x = {\ frac {x} {2}} [\ sin (\ log x) - \ cos (\ log x)] + C }

\int \cos(\log x)\;\mathrm {d} x={\frac {x}{2}}[\sin(\log x)+\cos(\log x)]+C

{\ displaystyle \ int \ cos (\ log x) \; \ mathrm {d} x = {\ frac {x} {2}} [\ sin (\ log x) + \ cos (\ log x)] + C }

{\ displaystyle \ int \ cos (\ log x) \; \ mathrm {d} x = {\ frac {x} {2}} [\ sin (\ log x) + \ cos (\ log x)] + C }

Bibliografie

Murray R. Spiegel, Manual de matematică , Etas Libri, 1974, p. 86.

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Tabelele integrale