جدول تكاملات دوال غير كسرية

analysis

تكاملات تتضمن $\sqrt {ax + b}$

$\int {\sqrt {ax + b} } \;dx\, = \,\frac{{2(ax + b)^{3/2} }}{{3a}} + C$

$\int {x\sqrt {ax + b} } \;dx\, = \,\frac{{2(3ax - 2b)}}{{15a^2 }}(ax + b)^{3/2} + C$

$\int {x^2 \sqrt {ax + b} } \;dx\, = \,\frac{{2(15a^2 x^2 - 12abx + 8b^2 )}}{{105a^3 }}(ax + b)^{3/2} + C$

$\int {\frac{{dx}}{{\sqrt {ax + b} }}} \, = \,\frac{{2\sqrt {ax + b} }}{a} + C$

$\int {\frac{{dx}}{{x\sqrt {ax + b} }}} \, = \,\frac{{ - 2}}{{\sqrt b }}\tanh ^{ - 1} \sqrt {\frac{{ax + b}}{b}} + C$

$\int {\frac{{\sqrt {ax + b} }}{x}} \,dx\: = \:2\left( {\sqrt {ax + b} - \sqrt b \tanh ^{ - 1} \sqrt {\frac{{ax + b}}{b}} } \right) + C$

$\int {\frac{{xdx}}{{\sqrt {ax + b} }}} \, = \,\frac{{2(ax - 2b)}}{{3a^2 }}\sqrt {ax + b} + C$

$\int {\frac{{x^2 dx}}{{\sqrt {ax + b} }}} \, = \,\frac{{2(3a^2 x^2 - 4abx + 8b^2 )}}{{15a^3 }}\sqrt {ax + b} + C$

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

$\int {x^n } \sqrt {ax + b} \,dx\: = \:\frac{2}{{2n + 1}}\left( {x^{n + 1} \sqrt {ax + b} + bx^n \sqrt {ax + b} - nb\int {x^{n - 1} } \sqrt {ax + b} \,dx} \right) + C$

تكاملات تتضمن $\sqrt {a^2 + x^2 }$

$\int {\sqrt {a^2 + x^2 } } \:dx = \frac{1}{2}\left( {x\sqrt {a^2 + x^2 } + a^2 \,\ln \left( {x + \sqrt {a^2 + x^2 } } \right)} \right) + C$

$\int {\frac{{dx}}{{\sqrt {a^2 + x^2 } }}} = \sinh ^{ - 1} \frac{x}{a} + C = \ln \left| {x + \sqrt {a^2 + x^2 } } \right| + C$

$\int {\frac{{x\,dx}}{{\sqrt {a^2 + x^2 } }}} = \sqrt {a^2 + x^2 } + C$

$\int {\left( {\sqrt {a^2 + x^2 } } \right)^3 } \:dx = \sqrt {a^2 + x^2 } \left( {\frac{{x^3 }}{4} + \frac{{5a^2 x}}{8}} \right) + \frac{{3a^4 }}{8}\ln \left( {x + \sqrt {a^2 + x^2 } } \right) + C$

$\int {\frac{{x\,dx}}{{\left( {\sqrt {a^2 + x^2 } } \right)^3 }}} = - \frac{1}{{\sqrt {a^2 + x^2 } }} + C$

$\int {\frac{{dx}}{{\left( {\sqrt {a^2 + x^2 } } \right)^3 }}} = \frac{x}{{a^2 \sqrt {a^2 + x^2 } }} + C$

$\int {\frac{{\sqrt {a^2 + x^2 } \:dx}}{x}} = \sqrt {a^2 + x^2 } - a\sinh ^{ - 1} \frac{a}{x} + C$

$\int {\frac{{x^2 \:dx}}{{\sqrt {a^2 + x^2 } }}} = \frac{x}{2}\sqrt {a^2 + x^2 } - \frac{{a^2 }}{2}\,\sinh ^{ - 1} \frac{x}{a} + C$

$\int {\frac{{dx}}{{x\sqrt {a^2 + x^2 } }}} = - \frac{1}{a}\,\sinh ^{ - 1} \frac{a}{x} + C = - \frac{1}{a}\ln \left| {\frac{{a + r}}{x}} \right| + C$

$\int x \left( {\sqrt {a^2 + x^2 } } \right)^{2n + 1} \:dx = \frac{{\left( {\sqrt {a^2 + x^2 } } \right)^{2n + 3} }}{{2n + 3}} + C$

$\int {x^3 } \left( {\sqrt {a^2 + x^2 } } \right)^{2n + 1} \:dx = \frac{{\left( {\sqrt {a^2 + x^2 } } \right)^{2n + 5} }}{{2n + 5}} - \frac{{a^3 \left( {\sqrt {a^2 + x^2 } } \right)^{2n + 3} }}{{2n + 3}} + C$

تكاملات تتضمن $\sqrt {x^2 - a^2 }$

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

$\int {\frac{{\:dx}}{{x\sqrt {x^2 - a^2 } }}} = \frac{1}{a}{\mathop{\rm arcsec}\nolimits} x\left| {\frac{x}{a}} \right| + C$

$\int {\frac{{\:dx}}{{x^2 \sqrt {x^2 - a^2 } }}} = \frac{{\sqrt {x^2 - a^2 } }}{{a^2 x}} + C$

$\int {\frac{{\:dx}}{{x^3 \sqrt {x^2 - a^2 } }}} = \frac{{\sqrt {x^2 - a^2 } }}{{2a^2 x^2 }} + \frac{1}{{2a^3 }}{\mathop{\rm arcsec}\nolimits} \left| {\frac{x}{a}} \right| + C$

$\int {\sqrt {x^2 - a^2 } dx} = \frac{1}{2}\left( {x\sqrt {x^2 - a^2 } - a^2 \ln \left| {x + \sqrt {x^2 - a^2 } } \right|} \right) + C$

$\int x \sqrt {x^2 - a^2 } \:dx = \frac{1}{3}\left( {\sqrt {x^2 - a^2 } } \right)^3 + C$

$\int {x^2 } \sqrt {x^2 - a^2 } \:dx = \frac{x}{4}\left( {\sqrt {x^2 - a^2 } } \right)^3 + \frac{{a^2 x}}{8}\sqrt {x^2 - a^2 } - \frac{{a^4 }}{8}\ln \left| {x + \sqrt {x^2 - a^2 } } \right| + C$

$\int {x^3 } \sqrt {x^2 - a^2 } \:dx = \frac{1}{5}\left( {\sqrt {x^2 - a^2 } } \right)^5 + \frac{{a^2 }}{3}\left( {\sqrt {x^2 - a^2 } } \right)^3 + C$

$\int {\frac{{\sqrt {x^2 - a^2 } \:dx}}{x}} = \sqrt {x^2 - a^2 } - a\cos ^{ - 1} \left| {\frac{a}{x}} \right| + C$

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

$\int {\frac{{\sqrt {x^2 - a^2 } \:dx}}{{x^3 }}} = \frac{{\sqrt {x^2 - a^2 } }}{{2x^2 }} + \frac{1}{{2a}}{\mathop{\rm arcsec}\nolimits} \left| {\frac{x}{a}} \right| + C$

$\int {\frac{{x\:dx}}{{\sqrt {x^2 - a^2 } }}} = \sqrt {x^2 - a^2 } + C$

$\int {\frac{{x\:dx}}{{\left( {\sqrt {x^2 - a^2 } } \right)^{2n + 1} }}} = \frac{{ - 1}}{{(2n - 1)\left( {\sqrt {x^2 - a^2 } } \right)^{2n - 1} }} + C$

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

$\int {\frac{{dx}}{{\left( {\sqrt {x^2 - a^2 } } \right)^3 }}} = \frac{{ - x}}{{a^2 \sqrt {x^2 - a^2 } }} + C$

$\int {\frac{{x\:dx}}{{\left( {\sqrt {x^2 - a^2 } } \right)^3 }}} = \frac{{ - 1}}{{\sqrt {x^2 - a^2 } }} + C$

$\int {\frac{{x^2 \:dx}}{{\left( {\sqrt {x^2 - a^2 } } \right)^3 }}} = \frac{{ - x}}{{\sqrt {x^2 - a^2 } }} + \ln \left| {\frac{{x + \sqrt {x^2 - a^2 } }}{a}} \right| + C$

$\int {\frac{{x^3 \:dx}}{{\left( {\sqrt {x^2 - a^2 } } \right)^3 }}} = \sqrt {x^2 - a^2 } - \frac{{a^2 }}{{\sqrt {x^2 - a^2 } }} + C$

$\int {\frac{{x^{2m} \:dx}}{{\left( {\sqrt {x^2 - a^2 } } \right)^{2n + 1} }}} = \frac{{( - 1)^{n - m} }}{{a^{2(n - m)} }}\sum\limits_{i = 0}^{n - m - 1} {\frac{{C(n - m - 1,i)x^{2(m + i) + 1} }}{{\left( {2(m + i) + 1} \right)\left( {\sqrt {x^2 - a^2 } } \right)^{2(m + i) + 1} }}} \quad {\rm{(}}n > m \ge 0{\rm{)}}$

تكاملات تتضمن $\sqrt {a^2 - x^2 }$

$\int {\sqrt {a^2 - x^2 } } \:dx = \frac{1}{2}\left( {x\sqrt {a^2 - x^2 } + a^2 \arcsin \frac{x}{a}} \right) + C\quad {\rm{(}}|x| \le |a|{\rm{)}}$

$\int {\sqrt {a^2 - x^2 } } \:dx = \frac{1}{2}\left( {x\sqrt {a^2 - x^2 } - {\mathop{\rm sgn}} x\,\cosh ^{ - 1} \left| {\frac{x}{a}} \right|} \right) + C\quad {\rm{(for }}|x| \ge |a|{\rm{)}}$

$\int x \sqrt {a^2 - x^2 } \:dx = - \frac{1}{3}\left( {\sqrt {a^2 - x^2 } } \right)^3 + C\quad {\rm{(}}|x| \le |a|{\rm{)}}$

$\int {x^2 } \sqrt {a^2 - x^2 } \:dx = \frac{{x\left( {\sqrt {a^2 - x^2 } } \right)^3 }}{4} + \frac{{a^2 x\sqrt {a^2 - x^2 } }}{8} + \frac{{a^4 }}{8}\arcsin \frac{x}{a} + C$

$\int {x^3 } \sqrt {a^2 - x^2 } \:dx = \frac{{\left( {\sqrt {a^2 - x^2 } } \right)^5 }}{5} - \frac{{a^2 \left( {\sqrt {a^2 - x^2 } } \right)^3 }}{3} + C$

$\int {\frac{{\sqrt {a^2 - x^2 } \:dx}}{x}} = \sqrt {a^2 - x^2 } - a\ln \left| {\frac{{a + \sqrt {a^2 - x^2 } }}{x}} \right|\quad {\rm{(}}|x| \le |a|{\rm{)}}$

$\int {\frac{{dx}}{{\sqrt {a^2 - x^2 } }}} = \arcsin \frac{x}{a}{\rm{ + C}}\quad {\rm{(}}|x| \le |a|{\rm{)}}$

$\int {\frac{{x\:dx}}{{\sqrt {a^2 - x^2 } }}} = \sqrt {a^2 - x^2 } + C\quad {\rm{(}}|x| \le |a|{\rm{)}}$

$\int {\frac{{x^2 \:dx}}{{\sqrt {a^2 - x^2 } }}} = \frac{1}{2}\left( { - x\sqrt {a^2 - x^2 } + a^2 \arcsin \frac{x}{a}} \right) + C\quad {\rm{(}}|x| \le |a|{\rm{)}}$

$\int {\frac{{x^3 \:dx}}{{\sqrt {a^2 - x^2 } }}} = \frac{{\left( {\sqrt {a^2 - x^2 } } \right)^3 }}{3} - a^2 \sqrt {a^2 - x^2 } + C\quad {\rm{(}}|x| \le |a|{\rm{)}}$

$\int {\frac{{dx}}{{\left( {\sqrt {a^2 - x^2 } } \right)^3 }}} = \frac{x}{{a^2 \sqrt {a^2 - x^2 } }}{\rm{ + C}}$

$\int {\frac{{xdx}}{{\left( {\sqrt {a^2 - x^2 } } \right)^3 }}} = \frac{1}{{\sqrt {a^2 - x^2 } }}{\rm{ + C}}$

$\int {\frac{{x^2 \:dx}}{{\left( {\sqrt {a^2 - x^2 } } \right)^3 }}} = \frac{x}{{\sqrt {a^2 - x^2 } }} - \arcsin \frac{x}{a} + C$

$\int {\frac{{x^3 \:dx}}{{\left( {\sqrt {a^2 - x^2 } } \right)^3 }}} = \sqrt {a^2 - x^2 } + \frac{{a^2 }}{{\sqrt {a^2 - x^2 } }} + C$

$\int {\frac{{dx}}{{x\sqrt {a^2 - x^2 } }}} = \frac{1}{a}\ln \left| {\frac{{a + \sqrt {a^2 - x^2 } }}{x}} \right| + C$

$\int {\frac{{dx}}{{x^2 \sqrt {a^2 - x^2 } }}} = \frac{{ - \sqrt {a^2 - x^2 } }}{{a^2 x}} + C$

$\int {\frac{{\:dx}}{{x^3 \sqrt {a^2 - x^2 } }}} = \frac{{ - \sqrt {a^2 - x^2 } }}{{2a^2 x^2 }} - \frac{1}{{2a^3 }}\ln \left| {\frac{{a + \sqrt {a^2 - x^2 } }}{x}} \right| + C$

تكاملات تتضمن $S = \sqrt {ax^2 + bx + c}$

$\begin{array}{*{20}c} {\int {\frac{{dx}}{R}} } \hfill & { = \frac{1}{{\sqrt a }}\ln \left| {2\sqrt a R + 2ax + b} \right|\quad {\rm{(for }}a > 0{\rm{)}}} \hfill \\ {} \hfill & { = \frac{1}{{\sqrt a }}\,\sinh ^{ - 1} \frac{{2ax + b}}{{\sqrt {4ac - b^2 } }}\quad {\rm{(for }}a > 0{\rm{, }}4ac - b^2 > 0{\rm{)}}} \hfill \\ {} \hfill & { = \frac{1}{{\sqrt a }}\ln |2ax + b|\quad {\rm{(for }}a > 0{\rm{, }}4ac - b^2 = 0{\rm{)}}} \hfill \\ {} \hfill & \begin{array}{l} = - \frac{1}{{\sqrt { - a} }}\arcsin \frac{{2ax + b}}{{\sqrt {b^2 - 4ac} }} \\ {\rm{(for }}a < 0{\rm{, }}4ac - b^2 < 0{\rm{, }}\left| {2ax + b} \right| < \sqrt {b^2 - 4ac} {\rm{)}} \\ \end{array} \hfill \\ \end{array}$

$\int {Rdx} = \frac{{(2ax + b)R}}{{4a}} + \frac{{4ac - b^2 }}{{8a}}\int {\frac{{dx}}{R}}$

$\int {\frac{{xdx}}{R}} = \frac{R}{a} - \frac{b}{{2a}}\int {\frac{{dx}}{R}}$

$\int {\frac{{x^2 dx}}{R}} = \frac{{(2ax - 3b)}}{{4a^2 }}R + \frac{{3b^2 - 4ac}}{{8a^2 }}\int {\frac{{dx}}{R}}$

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

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

$\int {\frac{{dx}}{{R^{2n + 1} }}} = \frac{2}{{(2n - 1)(4ac - b^2 )}}\left( {\frac{{2ax + b}}{{R^{2n - 1} }} + 4a(n - 1)\int {\frac{{dx}}{{R^{2n - 1} }}} } \right)$

$\int {\frac{x}{{R^{2n + 1} }}} \:dx = - \frac{1}{{(2n - 1)aR^{2n - 1} }} - \frac{b}{{2a}}\int {\frac{{dx}}{{R^{2n + 1} }}}$

التكاملات الكسرية

قدم بواسطة Anonymous في أربعاء, 07/30/2008 - 16:16

شكرا لكم عمل رائع ومميز مها خالد

رد

جمع بديع وفقكم الله وسدد

قدم بواسطة الرواء (لم يتم التحقق) في ثلاثاء, 11/04/2008 - 18:44

جمع بديع وفقكم الله وسدد خطاكم

رد

علِّق

اسمك:

بريد إلكتروني:

محتويات هذا الحقل سرية ولن تظهر للآخرين.

الصفحة الأولى:

الموضوع:

التعليق: *

وسوم html المسموح بها: <a> <b> <em> <strong> <cite> <code> <ul> <ol> <li> <dl> <dt> <dd> <div> <span> <p> <br> <br /> <blockquote> <h1> <h2> <h3> <h4> <h5> <h6> <img> <sub> <sup> <table> <tbody> <tr> <td> <hr>
LaTeX formulas are automatically converted into images.
Use [fn]...[/fn] (or <fn>...</fn>) to insert automatically numbered footnotes.
Use [# ...] to insert automatically numbered footnotes. Textile variant.
Web page addresses and e-mail addresses turn into links automatically. (Better URL filter.)
Images can be added to this post.

معلومات أكثر عن خيارات التنسيق

كلمة التحقق

This question is for testing whether you are a human visitor and to prevent automated spam submissions.