Integrales:<\/p>\n\n\n\n
Definidas<\/a><\/p>\n\n\n\n Indefinidas<\/a><\/p>\n<\/div>\n\n\n\n Referencia<\/em><\/strong>: Leon W Couch Ap\u00e9ndice p657, 658<\/p>\n\n\n\n Integrales Definidas <\/strong><\/p>\n\n\n\n Definici\u00f3n<\/p>\n\n\n \\int f(x) dx = \\lim_{\\Delta \\rightarrow 0} \\left( \\sum_{n} \\left[ f(n \\Delta x)\\right] \\Delta x \\right) <\/span>\n\n\n\n Cambio de variable. Sea v=u(x)<\/p>\n\n\n \\int_{a}^{b} f(x) dx = \\int_{u(a)}^{u(b)} \\left( \\left. \\frac{f(x)}{dv\/dx} \\right|_{x=u^{-1}(v)}\\right) dv <\/span>\n\n\n\n integraci\u00f3n por partes<\/p>\n\n\n \\int u dv = uv - \\int v du <\/span>\n\n\n\n Integrales Definidas<\/p>\n\n\n \\int_{0}^{\\infty} \\frac{x^{m-1}}{1+x^n} dx = \\frac{\\pi \/n}{sen(m\\pi\/n)}, \\text{ }n>m>0 <\/span>\n\n\n\n Referencia:<\/strong> Leon W Couch Ap\u00e9ndice p656<\/p>\n\n\n\n Integrales:<\/p>\n\n\n\n Definidas<\/a><\/p>\n\n\n\n Indefinidas<\/a><\/p>\n<\/div>\n\n\n\n Trigonom\u00e9tricas<\/p>\n\n\n \\int cos(x) dx = sen(x) <\/span>\n\n\n \\int sen(x) dx = -cos(x) <\/span>\n\n\n \\int x cos(x) dx = cos(x) + x sen(x) <\/span>\n\n\n \\int x sen(x) dx = sen(x) - x cos(x) <\/span>\n\n\n \\int x^2 cos(x) dx = 2x cos(x) + (x^2 -2) sen(x) <\/span>\n\n\n \\int x^2 sen(x) dx = 2x sen(x) - (x^2 -2) cos(x) <\/span>\n\n\n\n Exponenciales<\/p>\n\n\n \\int e^{ax} dx = \\frac{e^{ax}}{a} <\/span>\n\n\n \\int x e^{ax} dx = e^{ax} \\left( \\frac{x}{a} - \\frac{1}{a^2} \\right) <\/span>\n\n\n \\int x^2 e^{ax} dx = e^{ax} \\left( \\frac{x^2}{a} - \\frac{2x}{a^2} + \\frac{2}{a^3} \\right) <\/span>\n\n\n \\int x^3 e^{ax} dx = e^{ax} \\left( \\frac{x^3}{a} - \\frac{3x^2}{a^2} + \\frac{6x}{a^3} - \\frac{6}{a^4}\\right) <\/span>\n\n\n \\int e^{ax} sen(x) dx = \\frac{e^{ax}}{a^2 +1} (a sen(x) - cos(x)) <\/span>\n\n\n \\int e^{ax} cos(x) dx = \\frac{e^{ax}}{a^2 +1} (a cos(x) - sen(x)) <\/span>\n\n\n\n Integrales:<\/p>\n\n\n\n
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\n\n\n \\int_{0}^{\\infty} x^{\\alpha-1}e^{-x} dx = \\Gamma(\\alpha) , \\alpha > 0 <\/span>\n\n\n \\text{donde: }\\Gamma(\\alpha +1) = \\alpha \\Gamma(\\alpha), <\/span>\n\n\n \\Gamma (1) = 1, <\/span>\n\n\n \\Gamma [1\/2] = \\sqrt{\\pi}, <\/span>\n\n\n \\Gamma(n) = (n-1)! \\text{, si n es entero positivo } <\/span>\n\n\n\n
\n\n\n \\int_{0}^{\\infty} x^{2n} e^{-ax^2} dx =\\frac{1 \\cdot 3 \\cdot 5 \\cdot \\cdot \\cdot (2n-1)}{2^{n+1}a^{n}} \\sqrt{\\frac{\\pi}{a}} <\/span>\n\n\n \\int_{-\\infty}^{\\infty} e^{-a^2 x^2 + bx} dx =\\frac{\\sqrt{\\pi}}{a} e^{b^2\/(4a^2)}, a>0 <\/span>\n\n\n \\int_{0}^{\\infty} e^{-ax}cos(bx) dx = \\frac{a}{a^2+b^2}, a>0 <\/span>\n\n\n \\int_{0}^{\\infty} e^{-ax}sen(bx) dx = \\frac{b}{a^2+b^2}, a>0 <\/span>\n\n\n \\int_{0}^{\\infty} e^{-a^2x^2}cos(bx) dx = \\frac{\\sqrt{\\pi} e^{-b^2\/4a^2}}{2a}, a>0 <\/span>\n\n\n\n
\n\n\n \\int_{0}^{\\infty} x^{\\alpha-1}cos(bx) dx = <\/span>\n\n\n\\frac{\\Gamma(\\alpha)}{b^{\\alpha}} cos \\left(\\frac{1}{2}\\pi \\alpha \\right), <\/span>\n\n\n 0<\\alpha < 1, b >0 <\/span>\n\n\n\n
\n\n\n \\int_{0}^{\\infty} x^{\\alpha-1}sen(bx) dx = \\frac{\\Gamma(\\alpha)}{b^{\\alpha}} sen \\left(\\frac{1}{2}\\pi \\alpha \\right),<\/span>\n\n\n 0<|\\alpha| < 1, b >0 <\/span>\n\n\n\n
\n\n\n \\int_{0}^{\\infty} x e^{-ax^2} I_k(bx) dx = \\frac{1}{2a} e^{b^2\/4a}, <\/span>\n\n\n \\text{donde: } I_k(bx)=\\frac{1}{\\pi}\\int_{0}^{\\pi} e^{bx cos(\\theta)} cos(k\\theta) d\\theta <\/span>\n\n\n\n
\n\n\n \\int_{0}^{\\infty} \\frac{sen(x)}{x} dx = \\int_{0}^{\\infty} Sa(x) dx = \\frac{\\pi}{2} <\/span>\n\n\n \\int_{0}^{\\infty} \\left( \\frac{sen(x)}{x} \\right)^2 dx = \\int_{0}^{\\infty} Sa^2(x) dx = \\frac{\\pi}{2} <\/span>\n\n\n \\int_{-\\infty}^{\\infty} e^{\\pm j2 \\pi yx} dx = \\delta (y) <\/span>\n\n\n \\int_{0}^{\\infty}\\frac{cos(ax)}{b^2 + x^2}dx = \\frac{\\pi}{2b} e^{-ab}, a>0,b>0 <\/span>\n\n\n \\int_{0}^{\\infty}\\frac{x sen(ax)}{b^2 + x^2}dx = \\frac{\\pi}{2} e^{-ab}, a>0,b>0 <\/span>\n\n\n\n
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\n\n\n\n Integrales Indefinidas <\/strong><\/h2>\n\n\n \\int (a+bx)^n dx = \\frac{(a+bx)^{n+1}} {b(n+1)}, 0<n <\/span>\n\n\n \\int \\frac{dx}{a+bx} =\\frac{1}{b} ln|a+bx| <\/span>\n\n\n \\int \\frac{dx}{(a+bx)^n} = \\frac{-1}{(n-1)b(a+bx)^{n-1}} , 1<n <\/span>\n\n\n\n
\n\n\n \\int \\frac{dx}{(c+bc+ax^2)^n} = <\/span>\n\n\n= \\begin{cases} \\frac{2}{ \\sqrt{4ac-b^2}} tan^{-1}\\left(\\frac{2ax+b}{\\sqrt{4ac-b^2}}\\right) , & 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| , & b^{2} > 4ac \\\\ \\frac{-2}{\\sqrt{2ax+b}} , & b^{2}=4ac \\end{cases} <\/span>\n\n\n\n
\n\n\n \\int \\frac{x dx}{c+bx+ax^2} = <\/span>\n\n\n = \\frac{1}{2a} ln\\left| ax^2+bx+c \\right| - \\frac{b}{2a}\\int \\frac{dx}{c+bx+ax^2} <\/span>\n\n\n\n
\n\n\n \\int \\frac{dx}{a^2+b^2x^2} = \\frac{1}{ab} tan^{-1}\\left( \\frac{bx}{a} \\right) <\/span>\n\n\n \\int \\frac{x dx}{a^2+x^2} = \\frac{1}{2} ln( a^2+x^2 ) <\/span>\n\n\n\n
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