Category Archives: រូបមន្តអាំងតេក្រាលសំខាន់ៗ

រូបមន្ត និងទំរង់អាំងតេក្រាលមួយចំនួន

  • ទំរង់ស្តង់ដា
1.\int adx=ax+c
2.\int af(x)dx=a \int f(x)dx+c
3.\int udv=uv- \int vdu (អាំងតេក្រាលដោយផ្នែក)
4.\int u^{n}du= \displaystyle \frac{u^{n+1}}{n+1}+c

5.\int \dfrac{du}{u}=ln|u|+c
6.\int e^{u}du=e^{u}+c
7.\int a^{u}du= \int e^{ulna}du= \displaystyle \frac{e^{ulna}}{lna}= \displaystyle \frac{a^{u}}{lna}+c
8. \int sinudu=-cosu+c
9.\int cosudu=sinu+c
10.\int tanudu=ln|secu|=-ln|cosu|+c
11.\int cotudu=ln|sinu|+c
12.\int secudu=ln|secu+tanu|+c=ln tan \displaystyle \frac{u}{2}+ \frac{ \pi}{4}+c
13.\int csc udu=ln|csc u-cotu|=ln|tan \frac{u}{2}|+c
14.\int sec^{2}udu=tanu+c
15.\int csc^{2}udu=-csc u+c
16.\int sec u tanudu=secu+c
17.\int csc ucotudu=-csc u+c

18.\int \displaystyle \frac{du}{u^{2}+a^{2}}= \frac{1}{a}acrtg \frac{u}{a}+c

19.\int \displaystyle \frac{du}{u^{2}-a^{2}}= \frac{1}{2a}ln \displaystyle | \frac{u-a}{u+a}|+c

20.\int \displaystyle \frac{du}{a^{2}-u^{2}}= \displaystyle \frac{1}{2a} ln| \displaystyle \frac{a+u}{a-u}|+c= \displaystyle \frac{1}{a}acrtgh \frac{u}{a}+c

21.\int \displaystyle \frac{du}{ \sqrt{a^{2}-u^{2}}}=arcsin \frac{u}{a}+c

22.\int \displaystyle \frac{du}{ \sqrt{u^{2}+a^{2}}}=ln(u+ \sqrt{u^{2}+a^{2}})+c

23.\int \displaystyle \frac{du}{ \sqrt{u^{2}-a^{2}}}=ln|u+ \sqrt{u^{2}-a^{2}}|+c

24.\int \displaystyle \frac{du}{u \sqrt{u^{2}-a^{2}}}= \displaystyle \frac{1}{|a|}arcsec \displaystyle | \frac{u}{a}|+c

25.\int \displaystyle \frac{du}{u \sqrt{u^{2}+a^{2}}}=- \displaystyle \frac{1}{a}ln| \displaystyle \frac{a+ \sqrt{u^{2}+a^{2}}}{u}|+c

26.\int \displaystyle \frac{du}{u \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{1}{a}ln | \displaystyle \frac{a+ \sqrt{a^{2}-u^{2}}}{u}|+c

  • ទំរង់ជាប់ au+b

27.\int \displaystyle \frac{du}{au+b}= \displaystyle \frac{1}{a}ln|au+b|+c

28.\int \displaystyle \frac{udu}{au+b}= \displaystyle \frac{u}{a}- \displaystyle \frac{b}{a^{2}}ln|au+b|+c

29.\int \displaystyle \frac{u^{2}du}{au+b}= \displaystyle \frac{(au+b)^{2}}{2a^{3}}- \displaystyle \frac{2b(au+b)}{a^{3}}+ \displaystyle \frac{b^{2}}{a^{3}}ln|au+b|+c

30.\int \displaystyle \frac{du}{u(au+b)}= \displaystyle \frac{1}{b}ln| \displaystyle \frac{u}{au+b}|+c

31.\int \displaystyle \frac{du}{u^{2}(au+b)}=- \displaystyle \frac{1}{bu}+ \displaystyle \frac{a}{b^{2}}ln| \displaystyle \frac{au+b}{u}|+c

32.\int \displaystyle \frac{du}{(au+b)^{2}}= \displaystyle \frac{-1}{a(au+b)}+c

33.\int \displaystyle \frac{udu}{(au+b)^{2}}= \displaystyle \frac{b}{a^{2}(au+b)}+ \displaystyle \frac{1}{a^{2}}ln|au+b|+c

34.\int \displaystyle \frac{du}{u(au+b)^{2}}= \displaystyle \frac{1}{b(au+b)}+ \displaystyle \frac{1}{b^{2}}ln| \displaystyle \frac{u}{au+b}|+c

35.\int (au+b)^{n}du= \displaystyle \frac{(au+b)^{n+1}}{(n+1)a}+c

36.\int u(au+b)^{n}du= \displaystyle \frac{(au+b)^{n+2}}{(n+2)a^{2}}- \displaystyle \frac{b(au+b)^{n+1}}{(n+1)a^{2}}+c

37.\int u^{m}(au+b)^{n}du= \begin{Bmatrix} \displaystyle \frac{u^{m+1}(au+b)^{n}}{m+n+1}+ \displaystyle \frac{nb}{m+n+1} \int u^{m}(au+b)^{n-1}du \\ \displaystyle \frac{u^{m}(au+b)^{n+1}}{(m+n+1)a}- \displaystyle \frac{mb}{(m+n+1)a} \int u^{n-1}(au+b)^{n}du \\ \displaystyle \frac{-u^{m+1}(au+b)^{n+1}}{(n+1)b}+ \displaystyle \frac{(m+n+2)}{(n+1)b} \int u^{m}(au+b)^{n+1}du \end{Bmatrix}

  • ទំរង់ជាប់ \sqrt{au+b}
  • 38.\int \displaystyle \frac{du}{ \sqrt{au+b}}= \displaystyle \frac{2 \sqrt{au+b}}{a}+c

    39.\int \displaystyle \frac{udu}{ \sqrt{au+b}}= \displaystyle \frac{2(au-2b)}{3a^{2}} \sqrt{au+b}+c

    40.\int \displaystyle \frac{du}{u \sqrt{au+b}}= \begin{Bmatrix} \displaystyle \frac{1}{ \sqrt{b}}ln| \displaystyle \frac{ \sqrt{au+b}- \sqrt{b}}{ \sqrt{au+b}+ \sqrt{b}}|+c,b>0 \\ \displaystyle \frac{2}{ \sqrt{-b}}arctg \sqrt{ \displaystyle \frac{au+b}{-b}}+c,b<0 \end{Bmatrix}

    41.\int \sqrt{au+b}du= \displaystyle \frac{2 \sqrt{(au+b)^{3}}}{3a}+c

    42.\int u \sqrt{au+b}du= \displaystyle \frac{2(3au-2b)}{15a^{2}} \sqrt{(au+b)^{3}}+c

    43.\int \displaystyle \frac{ \sqrt{au+b}}{u}du=2 \sqrt{au+b}+b \int \displaystyle \frac{du}{u \sqrt{au+b}} (see 40)

  • រូបមន្តជាប់ u^{2}+a^{2}
  • 44.\int \displaystyle \frac{du}{u^{2}+a^{2}}= \displaystyle \frac{1}{a}arctg \displaystyle \frac{u}{a}+c

    45.\int \displaystyle \frac{udu}{u^{2}+a^{2}}= \displaystyle \frac{1}{2}ln(u^{2}+a^{2})+c

    46.\int \displaystyle \frac{u^{2}du}{u^{2}+a^{2}}=u-a arctg \displaystyle \frac{u}{a}+c

    47.\int \displaystyle \frac{du}{u(u^{2}+a^{2})}= \displaystyle \frac{1}{2a^{2}}ln( \displaystyle \frac{u^{2}}{u^{2}+a^{2}})+c

    48.\int \displaystyle \frac{du}{u^{2}(u^{2}+a^{2})}=- \displaystyle \frac{1}{a^{2}u}- \displaystyle \frac{1}{a^{3}}arctg \displaystyle \frac{u}{a}+c

    49.\int \displaystyle \frac{du}{(u^{2}+a^{2})^{n}}= \displaystyle \frac{u}{2(n-1)a^{2}(u^{2}+a^{2})^{n-1}}+ \displaystyle \frac{2n-3}{(2n-2)a^{2}} \int \displaystyle \frac{du}{(u^{2}+a^{2})^{n-1}}

    50.\int \displaystyle \frac{udu}{(u^{2}+a^{2})^{n}}= \displaystyle \frac{-1}{2(n-1)(u^{2}+a^{2})^{n-1}}+c

    51.\int \displaystyle \frac{du}{u(u^{2}+a^{2})^{n}}= \displaystyle \frac{1}{2(n-1)a^{2}(u^{2}+a^{2})^{n-1}}+ \displaystyle \frac{1}{a^{2}} \int \displaystyle \frac{du}{u(u^{2}+a^{2})^{n-1}}+c

  • រូបមន្តជាប់ u^{2}-a^{2},u^{2}>a^{2}
  • 52.\int \displaystyle \frac{du}{u^{2}-a^{2}}= \displaystyle \frac{1}{2a}ln| \displaystyle \frac{u-a}{u+a}|+c

    53.\int \displaystyle \frac{udu}{u^{2}-a^{2}}= \displaystyle \frac{1}{2}ln(u^{2}-a^{2})+c

    54.\int \displaystyle \frac{u^{2}du}{u^{2}-a^{2}}=u+ \displaystyle \frac{a}{2}ln| \displaystyle \frac{u-a}{u+a}|+c

    55.\int \displaystyle \frac{du}{u(u^{2}-a^{2})}= \displaystyle \frac{1}{2a^{2}}ln| \displaystyle \frac{u^{2}-a^{2}}{u^{2}}|+c

    56.\int \displaystyle \frac{du}{u^{2}(u^{2}-a^{2})}= \displaystyle \frac{1}{a^{2}u}+ \displaystyle \frac{1}{2a^{3}} ln| \displaystyle \frac{u-a}{u+a}|+c

    57.\int \displaystyle \frac{du}{(u^{2}-a^{2})^{2}}= \displaystyle \frac{-u}{2a^{2}(u^{2}-a^{2})}- \displaystyle \frac{1}{4a^{3}}ln| \displaystyle \frac{u-a}{u+a}|+c

    58.\int \displaystyle \frac{du}{(u^{2}-a^{2})^{n}}= \displaystyle \frac{-u}{2(n-1)a^{2}(u^{2}-a^{2})^{n-1}}- \displaystyle \frac{2n-3}{(2n-2)a^{2}} \int \displaystyle \frac{du}{(u^{2}-a^{2})^{n-1}}

    59.\int \displaystyle \frac{udu}{(u^{2}-a^{2})^{n}}= \displaystyle \frac{-1}{2(n-1)a^{2}(u^{2}-a^{2})^{n-1}}+c

    60.\int \displaystyle \frac{du}{u(u^{2}-a^{2})^{n}}= \displaystyle \frac{-1}{2(n-1)a^{2}(u^{2}-a^{2})^{n-1}}- \displaystyle \frac{1}{a^{2}} \int \displaystyle \frac{du}{u(u^{2}-a^{2})^{n-1}}

  • រូបមន្តជាប់ a^{2}-u^{2},u^{2}<a^{2}
  • 61.\int \displaystyle \frac{du}{a^{2}-u^{2}}= \displaystyle \frac{1}{2a}ln| \displaystyle \frac{a+u}{a-u}|+c

    62.\int \displaystyle \frac{udu}{a^{2}-u^{2}}=- \displaystyle \frac{1}{2}ln|a^{2}-u^{2}|+c

    63.\int \displaystyle \frac{u^{2}du}{a^{2}-u^{2}}=-u+ \displaystyle \frac{a}{2}ln| \displaystyle \frac{a+u}{a-u}|+c

    64.\int \displaystyle \frac{du}{u(a^{2}-u^{2})}= \displaystyle \frac{1}{2a^{2}}ln| \displaystyle \frac{u^{2}}{a^{2}-u^{2}}|+c

    65.\int \displaystyle \frac{du}{(a^{2}-u^{2})^{2}}= \displaystyle \frac{u}{2a^{2}(a^{2}-u^{2})}+ \displaystyle \frac{1}{4a^{3}}ln| \displaystyle \frac{a+u}{a-u}|+c

    66.\int \displaystyle \frac{udu}{(a^{2}-u^{2})^{2}}= \displaystyle \frac{1}{2(a^{2}-u^{2})}+c

  • រូបមន្តជាប់ \sqrt{u^{2}+a^{2}}
  • 67.\int \displaystyle \frac{du}{ \sqrt{u^{2}+a^{2}}}=ln(u+ \sqrt{u^{2}+a^{2}})+c=arcsin \displaystyle \frac{u}{|a|}+c

    68.\int \displaystyle \frac{udu}{ \sqrt{u^{2}+a^{2}}}= \sqrt{u^{2}+a^{2}}+c

    69.\int \displaystyle \frac{u^{2}du}{ \sqrt{u^{2}+a^{2}}}= \displaystyle \frac{u \sqrt{u^{2}+a^{2}}}{2}- \displaystyle \frac{a^{2}}{2}ln(u+ \sqrt{u^{2}+a^{2}})+c

    70.\int \displaystyle \frac{du}{u \sqrt{u^{2}+a^{2}}}=- \displaystyle \frac{1}{a}ln| \displaystyle \frac{a+ \sqrt{u^{2}+a^{2}}}{u}|+c

    71.\int \sqrt{u^{2}+a^{2}}du= \displaystyle \frac{u \sqrt{u^{2}+a^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}ln(u+ \sqrt{u^{2}+a^{2}})+c

    72.\int u \sqrt{u^{2}+a^{2}}du= \displaystyle \frac{(u^{2}+a^{2})^{ \frac{3}{2}}}{3}+c

    73.\int u^{2} \sqrt{u^{2}+a^{2}}du= \displaystyle \frac{u(u^{2}+a^{2})^{ \frac{3}{2}}}{4}- \displaystyle \frac{a^{2}u \sqrt{u^{2}+a^{2}}}{8}- \displaystyle \frac{a^{4}}{8}ln(u+ \sqrt{u^{2}+a^{2}})+c

    74.\int \displaystyle \frac{ \sqrt{u^{2}+a^{2}}}{u}du= \sqrt{u^{2}+a^{2}}-aln| \displaystyle \frac{a+ \sqrt{u^{2}+a^{2}}}{u}|+c

    75.\int \displaystyle \frac{ \sqrt{u^{2}+a^{2}}}{u^{2}}du=- \displaystyle \frac{ \sqrt{u^{2}+a^{2}}}{u}+ln(u+ \sqrt{u^{2}+a^{2}})+c

  • រូបមន្តជាប់ \sqrt{u^{2}-a^{2}}
  • 76.\int \displaystyle \frac{du}{ \sqrt{u^{2}-a^{2}}}=ln|u+ \sqrt{u^{2}-a^{2}|}+c

    77.\int \displaystyle \frac{udu}{ \sqrt{u^{2}-a^{2}}}= \sqrt{u^{2}-a^{2}}+c

    78.\int \displaystyle \frac{u^{2}du}{ \sqrt{u^{2}-a^{2}}}= \displaystyle \frac{u \sqrt{u^{2}-a^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}ln|u+ \sqrt{u^{2}-a^{2}}|+c

    79.\int \displaystyle \frac{du}{u \sqrt{u^{2}-a^{2}}}= \displaystyle \frac{1}{|a|}arcsec| \displaystyle \frac{u}{a}|+c

    80.\int \sqrt{u^{2}-a^{2}}du= \displaystyle \frac{u \sqrt{u^{2}-a^{2}}}{2}- \displaystyle \frac{a^{2}}{2}ln|u+ \sqrt{u^{2}-a^{2}}|+c

    81.\int u \sqrt{u^{2}-a^{2}}du= \displaystyle \frac{(u^{2}-a^{2})^{ \frac{3}{2}}}{3}+c

    82.\int u^{2} \sqrt{u^{2}-a^{2}}du= \displaystyle \frac{u(u^{2}-a^{2})^{ \frac{3}{2}}}{4}+ \displaystyle \frac{a^{2}u \sqrt{u^{2}-a^{2}}}{8}- \displaystyle \frac{a^{4}}{8}ln|u+ \sqrt{u^{2}-a^{2}}|+c

    83.\int \displaystyle \frac{ \sqrt{u^{2}-a^{2}}}{u}du= \sqrt{u^{2}-a^{2}}-|a|arcsec| \displaystyle \frac{u}{a}|+c

    84.\int \displaystyle \frac{ \sqrt{u^{2}-a^{2}}}{u^{2}}du=- \displaystyle \frac{ \sqrt{u^{2}-a^{2}}}{u}+ln|u+ \sqrt{u^{2}-a^{2}}|+c

    85.\int \displaystyle \frac{du}{(u^{2}-a^{2})^{ \frac{3}{2}}}=- \displaystyle \frac{u}{a^{2} \sqrt{u^{2}-a^{2}}}+c

  • រូបមន្តជាប់ \sqrt{a^{2}-u^{2}}
  • 86.\int \displaystyle \frac{du}{ \sqrt{a^{2}-u^{2}}}=arcsin \displaystyle \frac{u}{|a|}+c

    87.\int \displaystyle \frac{udu}{ \sqrt{a^{2}-u^{2}}}=- \sqrt{a^{2}-u^{2}}+c

    88.\int \displaystyle \frac{u^{2}du}{ \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{u \sqrt{a^{2}-u^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}arcsin \displaystyle \frac{u}{|a|}+c

    89.\int \displaystyle \frac{du}{u \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{1}{a}ln| \displaystyle \frac{a+ \sqrt{a^{2}-u^{2}}}{u}|+c

    90.\int \displaystyle \frac{du}{u^{2} \sqrt{a^{2}-u^{2}}}=- \displaystyle \frac{ \sqrt{a^{2}-u^{2}}}{a^{2}u}+c

    91.\int \sqrt{a^{2}-u^{2}}du= \displaystyle \frac{u \sqrt{a^{2}-u^{2}}}{2}+ \displaystyle \frac{a^{2}}{2}arcsin \displaystyle \frac{u}{|a|}+c

    92.\int u \sqrt{a^{2}-u^{2}}du=- \displaystyle \frac{(a^{2}-u^{2})^{ \frac{3}{2}}}{3}+c

    93.\int u^{2} \sqrt{a^{2}-u^{2}}du=- \displaystyle \frac{u(a^{2}-u^{2})^{ \frac{3}{2}}}{4}+ \displaystyle \frac{a^{2}u \sqrt{a^{2}-u^{2}}}{8}+ \displaystyle \frac{a^{4}}{8}arcsin \displaystyle \frac{u}{|a|}+c

    94.\int \displaystyle \frac{ \sqrt{a^{2}-u^{2}}}{u}du= \sqrt{a^{2}-u^{2}}-aln| \displaystyle \frac{a+ \sqrt{a^{2}-u^{2}}}{u}|+c

    95.\int \displaystyle \frac{ \sqrt{a^{2}-u^{2}}}{u^{2}}du=- \displaystyle \frac{ \sqrt{a^{2}-u^{2}}}{u}-arcsin \displaystyle \frac{u}{|a|}+c

  • អាំងតេក្រាលជាប់អនុគមន៍ត្រីកោណមាត្រ
  • 96.\int sinudu=- \displaystyle \frac{cosau}{a}+c

    97.\int u sinaudu= \displaystyle \frac{sinau}{a^{2}}- \frac{ucosau}{a}+c

    98.\int u^{2}sinaudu= \displaystyle \frac{2u}{a^{2}}sinau+( \frac{2}{a^{3}}- \frac{u^{2}}{a})cosau+c

    99.\int \displaystyle \frac{du}{sinau}= \frac{1}{a}ln(cscau-cotau)= \frac{1}{a}ln|tan \frac{au}{2}|+c

    100.\int sin^{2}audu= \displaystyle \frac{u}{2}- \frac{sin2au}{4a}+c

    101.\int usin^{2}audu= \displaystyle \frac{u^{2}}{4}- \frac{usin2au}{4a}- \frac{cos2au}{8a^{2}}+c

    102.\int \displaystyle \frac{du}{sin^{2}au}=- \displaystyle \frac{1}{a}cotau+c

    103.\int sinpusinqudu= \displaystyle \frac{sin(p-q)u}{2(p-q)}- \frac{sin(p+q)u}{2(p+q)}+c

    104.\int \displaystyle \frac{du}{1-sinau}= \displaystyle \frac{1}{a}tan( \frac{ \pi}{4}+ \frac{au}{2})+c

    105.\int \displaystyle \frac{udu}{1-sinau}= \displaystyle \frac{u}{a}tan( \frac{ \pi}{4}+ \frac{au}{2})+ \frac{2}{a^{2}}ln|sin( \frac{ \pi}{4}- \frac{au}{2})|+c

    106.\int \displaystyle \frac{du}{1+sinau}=- \displaystyle \frac{1}{a}tan( \frac{ \pi}{4}- \frac{au}{2})+c

    107.\int \displaystyle \frac{du}{p+qsinau}=- \begin{Bmatrix} \displaystyle \frac{2}{a \sqrt{p^{2}-q^{2}}}arctg \displaystyle \frac{ptan \frac{1}{2}au+q}{ \sqrt{p^{2}-q^{2}}}+c \\ \displaystyle \frac{1}{a \sqrt{p^{2}-q^{2}}}ln| \displaystyle \frac{ptan \frac{1}{2}au+q- \sqrt{p^{2}-q^{2}}}{ptan \frac{1}{2}au+q+ \sqrt{p^{2}-q^{2}}}|+c \end{Bmatrix}

    108.\int u^{m}sinaudu=- \displaystyle \frac{u^{m}cosau}{a}+ \frac{mu^{m-1}sinau}{a^{2}}- \frac{m(m-1)}{a^{2}} \int u^{m-2}sinaudu

    109.\int sin^{n}audu=- \displaystyle \frac{sin^{n-1}aucosau}{an}+ \frac{n-1}{n} \int sin^{n-2}audu

    110.\int \displaystyle \frac{du}{sin^{n}au}= \frac{-cosau}{a(n-1)sin^{n-1}au}+ \frac{n-2}{n-1} \int \frac{du}{sin^{n-2}au}

    111.\int cosaudu= \displaystyle \frac{sinau}{a}+c

    112.\int ucosaudu= \displaystyle \frac{cosau}{a^{2}}+ \frac{usinau}{a}+c

    113.\int u^{2}cosaudu= \displaystyle \frac{2u}{a}cosau+( \frac{u^{2}}{a}- \frac{2}{a^{3}})sinau+c

    114.\int \displaystyle \frac{du}{coau}= \frac{1}{a}ln|secau+tanau|= \frac{1}{a}ln|tan( \frac{ \pi}{4}+ \frac{au}{2})|+c

    115.\int cos^{2}audu= \displaystyle \frac{u}{2}+ \frac{sin2au}{4a}+c

    116.\int ucos^{2}audu= \displaystyle \frac{u^{2}}{4}+ \frac{usin2au}{4a}+ \frac{cos2ar}{8a^{2}}+c

    117.\int \displaystyle \frac{du}{cos^{2}au}= \frac{tanau}{a}+c

    118.\int cosqucospudu= \displaystyle \frac{sin(q-p)u}{2(q-p)}+ \frac{sin(q+p)u}{2(q+p)}+c

    119.\int \displaystyle \frac{du}{p+qcosau}= \begin{Bmatrix} \displaystyle \frac{2}{a \sqrt{p^{2}-q^{2}}}arctg[ \sqrt{(p-q)/(p+q)}tan \frac{1}{2}au]+c,|p|>|q| \\ \displaystyle \frac{1}{a \sqrt{q^{2}-p^{2}}}ln[ \frac{tan \frac{1}{2}au+ \sqrt{(q+p)/(q-p)}}{tan \frac{1}{2}au- \sqrt{(q+p)/q-p)}}]+c,|p|<|q| \end{Bmatrix}

    120.\int u^{m}cosaudu= \displaystyle \frac{u^{m}sinau}{a}+ \frac{mu^{m-1}}{a^{2}}cosau- \frac{m(m-1)}{a^{2}} \int u^{m-2}cosaudu

    121.\int cos^{n}audu= \displaystyle \frac{sinaucos^{n-1}au}{an}+ \frac{n-1}{n} \int cos^{n-2}audu

    122.\int \displaystyle \frac{du}{cos^{n}au}= \frac{sinau}{a(n-1)cos^{n-1}au}+ \frac{n-2}{n-1} \int \frac{du}{cos^{n-2}au}

    123.\int sinau cosaudu= \displaystyle \frac{sin^{2}au}{2a}+c

    124.\int sinpucosqudu=- \displaystyle \frac{cos(p-q)u}{2(p-q)}- \frac{cos(p+q)u}{2(p+q)}+c

    125.\int sin^{n}aucosaudu= \displaystyle \frac{sin^{n+1}au}{(n+1)a}+c

    126.\int cos^{n}ausinaudu=- \displaystyle \frac{cos^{n+1}au}{(n+1)a}+c

    127.\int sin^{2}aucos^{2}audu= \displaystyle \frac{u}{8}- \frac{sin4au}{32a}+c

    128.\int \displaystyle \frac{du}{sinaucosau}= \frac{1}{a}ln|tanau|+c

    129.\int \displaystyle \frac{du}{cosau(1+sinau}= \frac{1}{2a(1+sinau)}+ \frac{1}{2a}ln|tan( \frac{au}{2}+ \frac{ \pi}{4})|+c

    130.\int \displaystyle \frac{du}{sinau(1 \pm cosau)}= \pm \frac{1}{2a(1 \pm cosau)}+ \frac{1}{2a}ln|tan \frac{au}{2}|+c

    131.\int \displaystyle \frac{du}{sinau \pm cosau}= \frac{1}{a \sqrt{2}}ln|tan( \frac{au}{2} \pm \frac{ \pi}{8})|+c

    132.\int \displaystyle \frac{sinaudu}{sinau \pm cosau}= \frac{u}{2} \mp \frac{1}{2a}ln|sinau \pm cosau|+c

    133.\int \displaystyle \frac{cosaudu}{sinau \pm cosau}= \pm [ \frac{u}{2}+ \frac{1}{2a}ln|sinau \pm cosau|]+c

    134.\int \displaystyle \frac{sinaudu}{p+qcosau}=- \frac{1}{aq}ln|p+qcosau|+c

    135.\int \displaystyle \frac{cosaudu}{p+qsinau}= \frac{1}{aq}ln|p+qsinau|+c

    136.\int sin^{m}aucos^{n}audu=\displaystyle \begin{Bmatrix} - \displaystyle \frac{sin^{m-1}aucos^{n+1}au}{a(n+m)}+ \frac{m-1}{m+n} \int sin^{m-2}aucos^{n}audu \\ \displaystyle \frac{sin^{m+1}aucos^{n-1}au}{a(m+n)}+ \frac{n-1}{m+n} \int sin^{m}aucos^{n-2}audu \end{Bmatrix}

    137.\int tanaudu=- \displaystyle \frac{1}{a}ln|cosau|= \frac{1}{a}ln|secau|+c

    138.\int tan^{2}audu= \displaystyle \frac{tanau}{a}-u+c

    139.\int tan^{n}ausec^{2}audu= \displaystyle \frac{tan^{n-1}au}{(n+1)a}+c

    140.\int tan^{n}audu= \displaystyle \frac{tan^{n-1}au}{(a-1)a}- \int tan^{n-2}audu

    141.\int cotaudu= \displaystyle \frac{1}{a}ln|sinau|+c

    142.\int cot^{2}audu=- \displaystyle \frac{cotau}{a}-u+c

    143.\int cot^{n}aucsc^{2}audu=- \displaystyle \frac{cot^{n-1}au}{(n+1)a}+c

    144.\int cot^{n}audu=- \displaystyle \frac{cot^{n-1}au}{(n-1)a}- \int cot^{n-2}audu

    145.\int sec^{2}audu= \displaystyle \frac{tanau}{a}+c

    146.\int secaudu= \displaystyle \frac{1}{a}ln|secau+tanau|= \frac{1}{a}ln|tan( \frac{au}{2}+ \frac{ \pi}{4})|+c

    147.\int sec^{3}audu= \displaystyle \frac{secautanau}{2a}+ \frac{1}{2a}ln|secau+tanau|+c

    148.\int sec^{n}autanaudu= \displaystyle \frac{sec^{n}au}{na}+c

    149.\int sec^{n}audu= \displaystyle \frac{sec^{n-2}autanau}{a(n-1)}+ \frac{n-2}{n-1} \int sec^{n-2}audu

    150.\int cscaudu= \displaystyle \frac{1}{a}ln|cscau-cotau|= \frac{1}{a}ln|tan \frac{au}{2}|+c

    151.\int csc^{2}audu=- \displaystyle \frac{cotau}{a}+c

    152.\int csc^{n}aucotaudu=- \displaystyle \frac{csc^{n}au}{an}+c

    153.\int csc^{n}audu=- \displaystyle \frac{csc^{n-2}aucotau}{a(n-1)}+ \frac{n-2}{n-1} \int csc^{n-2}audu

  • អាំងតេក្រាលជាប់​អនុគមន៍ច្រាសត្រីកោណមាត្រ
  • 154.\int  arcsin \displaystyle \frac{u}{a}du=uarcsin \frac{a}{u}+ \sqrt{a^{2}-u^{2}}+c

    155.\int uarcsin \displaystyle \frac{u}{a}du=( \frac{u^{2}}{2}- \frac{a^{2}}{4})arcsin \frac{u}{a}+ \frac{u \sqrt{a^{2}-u^{2}}}{4}+c

    156.\int arccos \displaystyle \frac{u}{a}du=uarccos \frac{u}{a}- \sqrt{a^{2}-u^{2}}+c

    157.\int uarccos \displaystyle \frac{u}{a}du=( \frac{u^{2}}{2}- \frac{a^{2}}{4})arccos \frac{u}{a}- \frac{u \sqrt{a^{2}-u^{2}}}{4}+c

    158.\int arctg \displaystyle \frac{u}{a}du=uacrtg \frac{u}{a}- \frac{a}{2}ln(u^{2}+a^{2})+c

    159.\int uarctg \displaystyle \frac{u}{a}du= \frac{1}{2}(u^{2}+a^{2})acrtg \frac{u}{a}- \frac{au}{2}+c

    160.\int u^{m}acrsin \displaystyle \frac{u}{a}du= \frac{u^{m+1}}{m+1}arcsin \frac{u}{a}- \frac{1}{m+1} \int \frac{u^{m+1}}{ \sqrt{a^{2}-u^{2}}}du

    161.\int u^{m}arccos \displaystyle \frac{u}{a}du= \frac{u^{m+1}}{m+1}arccos \frac{u}{a}+ \frac{1}{m+1} \int \frac{u^{m+1}}{ \sqrt{a^{2}-u^{2}}}du

    162.\int u^{m}arctg \displaystyle \frac{u}{a}du= \frac{u^{m+1}}{m+1}arctg \frac{u}{a}- \frac{a}{m+1} \int \frac{u^{m+1}}{u^{2}+a^{2}}du

  • អាំងតេក្រាលជាប់ e^{au}
  • 163.\int e^{au}du= \displaystyle \frac{e^{au}}{a}+c

    164.\int ue^{au}du= \displaystyle \frac{e^{au}}{a}(u- \frac{1}{a}+c

    165.\int u^{2}e^{au}du= \displaystyle \frac{e^{au}}{a}(u^{2}- \frac{2u}{a}+ \frac{2}{a^{2}}+c

    166.\int u^{n}e^{au}du= \displaystyle \frac{u^{n}e^{au}}{a}- \frac{n}{a} \int u^{n-1}e^{au}du= \frac{e^{au}}{a}[u^{n}- \frac{nu^{n-1}}{a}+ \frac{n(n-1)u^{n-2}}{a^{2}}-...+ \frac{(-1)^{n}n!}{a^{n}}],បើn ជាចំនួនគត់វិជ្ជមាន

    167.\int \displaystyle \frac{du}{p-qe^{au}}= \frac{u}{p}- \frac{1}{ap}ln|p+qe^{au}|+c

    168.\int e^{au}sinbudu= \displaystyle \frac{e^{au}(asinbu-bcosbu)}{a^{2}+b^{2}}+c

    169.\int e^{au}cosbudu= \displaystyle \frac{e^{au}(acosbu+bsinbu)}{a^{2}+b^{2}}+c

    170.\int ue^{au}sinbudu= \displaystyle \frac{ue^{au}(asinbu-bcosbu)}{a^{2}+b^{2}}- \frac{e^{au}[(a^{2}-b^{2})sinbu-2abcosbu]}{(a^{2}-b^{2})^{2}}+c

    171.\int ue^{au}cosbudu= \displaystyle \frac{ue^{au}(acosbu+bsinbu)}{a^{2}+b^{2}}- \frac{e^{au}[(a^{2}-b^{2})cosbu+2absinbu]}{(a^{2}-b^{2})^{2}}+c

    172.\int e^{au}sin^{n}budu= \displaystyle \frac{e^{au}sin^{n-1}bu}{a^{2}+n^{2}b^{2}}(asinbu-nbcosbu)+ \frac{n(n-1)b^{2}}{a^{2}+n^{2}b^{2}} \int e^{au}sin^{n-2}budu

    173.\int e^{au}cos^{n}budu= \displaystyle \frac{e^{au}cos^{n-1}bu}{a^{2}+n^{2}b^{2}}(acosbu+nbsinbu)+ \frac{n(n-1)b^{2}}{a^{2}+n^{2}b^{2}} \int e^{au}cos^{n-2}budu

  • អាំងតេក្រាលជាប់ lnu
  • 174.\int lnudu=ulnu-u+c

    175.\int ulnudu= \displaystyle \frac{u^{2}}{2}(lnu- \frac{1}{2})+c

    176.\int u^{m}lnudu= \displaystyle \frac{u^{m+1}}{m+1}(lnu- \frac{1}{m+1}

    177.\int \displaystyle \frac{lnu}{u}du= \frac{1}{2}ln^{2}u+c

    178.\int \displaystyle \frac{ln^{n}udu}{u}= \frac{ln^{n+1}u}{n+1}+c, បើ​ n \neq -1

    179.\int \displaystyle \frac{du}{ulnu}=ln|lnu|+c

    180.\int ln^{n}udu=uln^{n}u-n \int ln^{n-1}udu, n \neq -1

    181.\int u^{m}ln^{n}udu= \displaystyle \frac{u^{m+1}ln^{n}u}{m+1}- \frac{n}{m+1} \int u^{m}ln{n-1}udu,(m,n \neq -1)

    182.\int ln(u^{2}+a^{2})du=uln(u^{2}+a^{2})-2u+2aarctg \frac{u}{a}+c

    183.\int ln|u^{2}+a^{2}|du=uln|u^{2}+a^{2}|-su+aln| \dfrac{u+a}{u-a}|+c

  • អាំងតេក្រាលជាប់អនុគមន៍អីប៉ែបូល
  • 184.\int sinhaudu= \dfrac{coshau}{a}+c

    185.\int usinhaudu= \dfrac{ucoshau}{a}- \dfrac{sinhau}{a^{2}}+c

    186.\int \dfrac{sinhau}{a}+c

    187.\int ucoshaudu= \dfrac{usinhau}{a}- \dfrac{coshau}{a^{2}}+c

    188. \int cosh^{2}audu= \dfrac{u}{2}+ \dfrac{sinhaucoshau}{2a}+c

    189.\int sinh^{2}audu= \dfrac{sinhaucoshau}{2a}- \dfrac{u}{2}+c

    190.\int sinh^{n}audu= \dfrac{sinh^{n-1}aucoshau}{an}- \dfrac{n-1}{n} \int sinh^{n-2}audu

    191.\int cosh^{n}audu= \dfrac{cosh^{n-1}ausinhau}{an}- \dfrac{n-1}{n} \int cosh^{n-2}audu

    192.\int sinhaucoshaudu= \dfrac{sinh^{2}au}{2a}+c

    193.\int sinhpu coshqudu= \dfrac{cosh(p+q)u}{2(p+q)}+ \dfrac{cosh(p-q)u}{2(p-q)}+c

    194.\int tanhaudu= \dfrac{1}{a}lncoshau+c

    195.\int tanh^{2}audu=u- \dfrac{tanhau}{a}+c

    196.\int tanh^{n}audu= \dfrac{-tanh^{n-1}au}{a(n-1)}+ \int tanh^{n-2}audu

    197.\int cothaudu= \dfrac{1}{a}ln|sinhau|+c

    198.\int coth^{2}audu=u- \dfrac{cothau}{a}+c

    199.\int sechaudu= \dfrac{2}{a}arctge^{au}+c

    200.\int sech^{2}audu= \dfrac{tanhau}{a}

    201.\int sech^{n}audu= \dfrac{sech^{n-2}autanhau}{a(n-1)}+ \dfrac{n-2}{n-1} \int sech^{n-2}audu

    202.\int cschaudu= \dfrac{1}{a}ln|tanh \dfrac{au}{2}|+c

    203.\int csch^{2}audu=- \dfrac{cothau}{a}+c

    204.\int sechu tanhudu= -sechu+c

    205.\int cschu coshudu=-cschu+c

  • អាំងតេក្រាលកំនត់មួយចំនួន
  • 206.\int_{0}^{ \infty} \dfrac{dx}{x^{2}+a^{2}}= \dfrac{ \pi}{2a}

    207.\int_{0}^{ \infty} \dfrac{x^{p-1}}{1+x}dx= \dfrac{ \pi}{sinp \pi}

    208.\int_{0}^{a} \dfrac{dx}{ \sqrt{a^{2}-x^{2}}}= \dfrac{ \pi}{2}

    209.\int_{0}^{a} sqrt{a^{2}-x^{2}}dx= \dfrac{ \pi a^{2}}{4}

    210.\int_{0}^{ \pi} sinmxsinnxdx= \begin{Bmatrix} 0, ( m,n, integers , m \neq n) \\ \dfrac{ \pi}{2}, ( m,n, integers , m=n) \end{Bmatrix}

    211.\int_{0}^{ \pi} cosmxcosnxdx= \begin{Bmatrix} o, ( m,n, integers , m \neq n) \\ \dfrac{ \pi}{2}, ( m,n, integers ,m=n) \end{Bmatrix}

    212.\int_{0}^{ \pi} sinmxcosnxdx= \begin{Bmatrix} o,( m,n,intergers ,m+n = even) \\ \dfrac{2m}{(m^{2}-n^{2}},(if m,n, integers , m+n = odd) \end{Bmatrix}

    213.\int_{o}^{ \frac{ \pi}{2}} sin^{2}xdx= \int_{0}^{ \frac{ \pi}{2}} cos^{2}xdx= \dfrac{ \pi}{4}

    214.\int_{0}^{ \infty} e^{-ax}cosbxdx= \dfrac{a}{a^{2}+b^{2}}

    215.\int_{0}^{ \infty} e^{-ax}sinbxdx= \dfrac{b}{a^{2}+b^{2}}

    216.\int_{0}^{ \infty} e^{-a^{2}x^{2}}dx= \dfrac{ \sqrt{ \pi}}{2a}

    217.\int_{0}^{ \frac{ \pi}{2}} sin^{2m}xdx= \int_{0}^{ \frac{ \pi}{2}} cos^{2m}xdx= \dfrac{1.3.5.....(2m-1)}{2.4.6.....2m} \dfrac{ \pi}{2}, m=1,2,3....

    218.\int_{0}^{ \frac{ \pi}{2}} sin^{2m+1}xdx= \int_{0}^{ \frac{ \pi}{2}} cos^{2m+1}xdx= \dfrac{2.4.6.....2m}{1.3.5.....(2m+1)}, m=1,2,3,....

    219.\int_{0}^{ \infty} \dfrac{e^{-x}}{ \sqrt{x}}dx= \sqrt{ \pi}

    220.\int_{0}^{1} x^{m}(lnx)^{n}dx= \dfrac{(-1)^{n}n!}{(m+1)^{n+1}}, m \neq -1

    Advertisements