CÁLCULO

INTEGRALES

 función absolutamente integrable, tal como indica su nombre, es una función cuyo valor absoluto es integrable, lo que significa que la integral del valor absoluto en todo el dominio es finita.

Para una función de variable real, tal que

${\displaystyle \int |f(x)|dx=\int f^{+}(x)dx+\int f^{-}(x)dx}$

donde

${\displaystyle f^{+}(x)=\max(f(x),0),\ \ \ f^{-}(x)=\max(-f(x),0),}$

tanto ${\displaystyle \int f^{+}(x)dx}$ como ${\displaystyle \int f^{-}(x)dx}$ deben ser finitos. En la integral de Lebesgue, este es exactamente el requisito para que f se considere integrable (con la integral que entonces equivale a ${\displaystyle \int f^{+}(x)dx-\int f^{-}(x)dx}$), de modo que, de hecho, "absolutamente integrable" significa lo mismo que "integrable según Lebesgue".

Lo mismo ocurre con una función de valores complejos. Definiendo

${\displaystyle f^{+}(x)=\max(\Re f(x),0)}$

${\displaystyle f^{-}(x)=\max(-\Re f(x),0)}$

${\displaystyle f^{+i}(x)=\max(\Im f(x),0)}$

${\displaystyle f^{-i}(x)=\max(-\Im f(x),0)}$

donde ${\displaystyle \Re f(x)}$ y ${\displaystyle \Im f(x)}$ son las partes real e imaginaria de ${\displaystyle f(x)}$. Entonces

${\displaystyle |f(x)|\leq f^{+}(x)+f^{-}(x)+f^{+i}(x)+f^{-i}(x)\leq {\sqrt {2}}|f(x)|}$

así que

${\displaystyle \int |f(x)|dx\leq \int f^{+}(x)dx+\int f^{-}(x)dx+\int f^{+i}(x)dx+\int f^{-i}(x)dx\leq {\sqrt {2}}\int |f(x)|dx}$

Esto muestra que la suma de las cuatro integrales es finita si y solo si la integral del valor absoluto es finita, y la función es integrable según Lebesgue solo si las cuatro integrales son finitas. Por lo tanto, tener una integral finita del valor absoluto equivale a las condiciones para que la función sea "integrable según Lebesgue".

lista de integrales de funciones exponenciales (Agréguese a cada integral una constante arbitraria).

${\displaystyle \int e^{cx}\;dx={\frac {1}{c}}e^{cx}}$

${\displaystyle \int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)}$

${\displaystyle \int x^{2}e^{cx}\;dx=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)}$

${\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx}$

${\displaystyle \int {\frac {e^{cx}\;dx}{x}}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}}$

${\displaystyle \int {\frac {e^{cx}\;dx}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}dx}{x^{n-1}}}\right)\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}$

${\displaystyle \int e^{cx}\ln x\;dx={\frac {1}{c}}\left(e^{cx}\ln |x|-\int {\frac {e^{cx}dx}{x}}\right)}$

${\displaystyle \int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}$

${\displaystyle \int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}$

${\displaystyle \int e^{cx}\sin ^{n}x\;dx={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;dx}$

${\displaystyle \int e^{cx}\cos ^{n}x\;dx={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;dx}$

${\displaystyle \int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2\sigma }}\left(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {(}}2)}}\right)}$

${\displaystyle \int _{-\infty }^{\infty }e^{-\alpha x^{2}}dx={\sqrt {\frac {\pi }{\alpha }}}=I(\alpha )}$

${\displaystyle \int _{-\infty }^{\infty }x^{2n}e^{-\alpha x^{2}}dx=(-1)^{n}{\frac {d^{n}}{d\alpha ^{n}}}\ I(\alpha )\ {\xrightarrow {\mbox{si n = 1}}}\ ={\frac {\sqrt {\pi }}{2\alpha ^{3/2}}}}$

 lista de integrales de funciones hiperbólicas.

${\displaystyle \int \sinh cx\,dx={\frac {1}{c}}\cosh cx}$

${\displaystyle \int \cosh cx\,dx={\frac {1}{c}}\sinh cx}$

${\displaystyle \int \sinh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}}$

${\displaystyle \int \cosh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}}$

${\displaystyle \int \sinh ^{n}cx\,dx={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,dx\qquad {\mbox{(para }}n>0{\mbox{)}}}$

también: ${\displaystyle \int \sinh ^{n}cx\,dx={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,dx\qquad {\mbox{(para }}n<0 -1="" annotation="" mbox="" n="" neq="">$

${\displaystyle \int \cosh ^{n}cx\,dx={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,dx\qquad {\mbox{(para }}n>0{\mbox{)}}}$

también: ${\displaystyle \int \cosh ^{n}cx\,dx=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx+{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,dx\qquad {\mbox{(para }}n<0 -1="" annotation="" mbox="" n="" neq="">$

${\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|}$

también: ${\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|}$

también: ${\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|}$

también: ${\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|}$

${\displaystyle \int {\frac {dx}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}}$

${\displaystyle \int {\frac {dx}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}cx}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}cx}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx={\frac {\cosh ^{n-1}cx}{c(n-m)\sinh ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m}cx}}dx\qquad {\mbox{(para }}m\neq n{\mbox{)}}}$

también: ${\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n+1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{(para }}m\neq 1{\mbox{)}}}$

también: ${\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n-1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{(para }}m\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m-1}cx}{c(m-n)\cosh ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n}cx}}dx\qquad {\mbox{(para }}m\neq n{\mbox{)}}}$

también: ${\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m+1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}$

también: ${\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx=-{\frac {\sinh ^{m-1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$

${\displaystyle \int x\sinh cx\,dx={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx}$

${\displaystyle \int x\cosh cx\,dx={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx}$

${\displaystyle \int \tanh cx\,dx={\frac {1}{c}}\ln |\cosh cx|}$

${\displaystyle \int \coth cx\,dx={\frac {1}{c}}\ln |\sinh cx|}$

${\displaystyle \int \tanh ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}$

${\displaystyle \int \coth ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}$

${\displaystyle \int \sinh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(para }}b^{2}\neq c^{2}{\mbox{)}}}$

${\displaystyle \int \cosh bx\cosh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(para }}b^{2}\neq c^{2}{\mbox{)}}}$

${\displaystyle \int \cosh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(para }}b^{2}\neq c^{2}{\mbox{)}}}$

${\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)}$

${\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)}$

${\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)}$

 lista de integrales de funciones racionales.

${\displaystyle \int (c)dx=c{x}}$

${\displaystyle \int (x)dx={\frac {x^{2}}{2}}}$

${\displaystyle \int (ax^{k}+b)^{n}dx={\frac {1}{a}}\int (ax^{k}+b)^{n}adx={\frac {1}{a}}{\frac {(ax^{k}+b)^{n+1}}{(n+1)kx}}\qquad {\mbox{(para }}n\neq -1{\mbox{)}}}$

${\displaystyle \int (ax+b)^{n}dx={\frac {1}{a}}\int (ax+b)^{n}adx={\frac {1}{a}}{\frac {(ax+b)^{n+1}}{n+1}}\qquad {\mbox{(para }}n\neq -1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{ax+b}}={\frac {1}{a}}\int {(ax+b)^{-1}}adx={\frac {1}{a}}\ln \left|ax+b\right|}$

${\displaystyle \int x(ax+b)^{n}dx={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\mbox{(para }}n\not \in \{-1,-2\}{\mbox{)}}}$

${\displaystyle \int {\frac {x\;dx}{ax+b}}={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|}$

${\displaystyle \int {\frac {x\;dx}{(ax+b)^{2}}}={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|}$

${\displaystyle \int {\frac {x\;dx}{(ax+b)^{n}}}={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}\qquad {\mbox{(para }}n\not \in \{1,2\}{\mbox{)}}}$

${\displaystyle \int {\frac {x^{2}\;dx}{ax+b}}={\frac {1}{a^{3}}}\left({\frac {(ax+b)^{2}}{2}}-2b(ax+b)+b^{2}\ln \left|ax+b\right|\right)}$

${\displaystyle \int {\frac {x^{2}\;dx}{(ax+b)^{2}}}={\frac {1}{a^{3}}}\left(ax+b-2b\ln \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)}$

${\displaystyle \int {\frac {x^{2}\;dx}{(ax+b)^{3}}}={\frac {1}{a^{3}}}\left(\ln \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right)}$

${\displaystyle \int {\frac {x^{2}\;dx}{(ax+b)^{n}}}={\frac {1}{a^{3}}}\left(-{\frac {1}{(n-3)(ax+b)^{n-3}}}+{\frac {2b}{(n-2)(a+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right)\qquad {\mbox{(para }}n\not \in \{1,2,3\}{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{x(ax+b)}}=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}$

${\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}$

${\displaystyle \int {\frac {dx}{x^{2}(ax+b)^{2}}}=-a\left({\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2}x}}-{\frac {2}{b^{3}}}\ln \left|{\frac {ax+b}{x}}\right|\right)}$

${\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}}$

${\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {artanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}\qquad {\mbox{(para }}|x|<|a|{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arcoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}\qquad {\mbox{(para }}|x|>|a|{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(para }}4ac-b^{2}>0{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|\qquad {\mbox{(for }}4ac-b^{2}<0 annotation="" mbox="">$

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

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

${\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(para }}4ac-b^{2}<0 annotation="" mbox="">$

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

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

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