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integrals and derivatives

integrals and derivatives

Last update 

assume (a) is a constant. S) for integrals D) for derivatives

Items (72)

  • S)x^-1 dx

    ln|x| + C

  • S)sinhax

    1/a *coshax + c

  • S)coshax

    1/a *sinhax + c

  • S)tanhax

    1/a *ln| coshax | + c

  • S)cschx

    1/a *ln|tanh(ax/2)|+c =1/a * ln|cschax-cothax|

  • S)sechx

    2/a*tan-1(e^ax)+c=1/a*tan-1(sinhax)+c=1/a*sin-1(tanhax)

  • S)cothx

    ln| sinhx | + c

  • D)tanhx

    sech2x

  • D)sechx

    -tanhx∙sechx

  • D)cschx

    -cothx∙cschx

  • D)cothx

    -csch2x

  • D)arcsinhx=sinh-1x

    1/√(1+x^2)

  • D)arccoshx=cosh-1x

    1/√(x^2-1)

  • D)arctanhx=tanh-1x

    1/(1-x^2)

  • D)arccothx=coth-1x

    -1/(x^2-1)

  • D)arcsechx=sech-1x

    -1/(x∙√(1-x^2))

  • D)arccschx=csch-1x

    -1/(|x|∙√(1+x^2))

  • D)sinx

    cosx

  • D)cosx

    -sinx

  • D)sin^2x

    2∙sinx∙cosx = sin2x

  • D)cos^2x

    -2∙sinx∙cosx = - sin2x

  • D)tanx

    sec^2x=1/(cos^2x) = 1+tan^2x

  • D)cotx

    -csc^2x=-1/(sin^2x) = -1-cot^2x

  • D)secx

    secx∙tanx

  • D)cscx

    -cscx∙cotx

  • D)arcsinx = sin-1x

    1/√(1-x^2)

  • D)arccosx = cos-1x

    -1/√(1-x^2)

  • D)arctanx = tan-1x

    1/(1+x^2)

  • D)arccotx = cot-1x

    -1/(1+x^2)

  • D)arcsecx = sec-1x

    1/(|x|∙√(x^2-1))

  • D)arccscx = csc-1x

    -1/(|x|∙√(x^2-1))

  • D)e^x

    e^x

  • D)e^(c∙x)

    c∙e^(c∙x)

  • D)x^x

    x^x(1+ln(x))

  • D)log(x), where the base is 10

    1/(x∙ln(10))

  • D)loga(x), where the base is a

    1/(x∙ln(a))

  • D)ln(x)

    1/x

  • D)f^ g  ,f and g are both functions

    f ^g(g '∙ln(f)+(g/f)∙f ')

  • D)a^x

    ln(a)*a^x

  • S)sinx

    -cosx+c

  • S)cosx

    sinx+c

  • S)sin2x

    x/2-sin(2x)/4+c=(x-sinx∙cosx)/2+c

  • S)cos^2x

    x/2+sin(2x)/4+c=(x+sinx*cosx)/2+c

  • S)tanx

    -ln|cosx|+c=ln|secx|

  • S)cotx

    ln|sinx|+c

  • S)secx

    ln|secx+tanx|+c

  • S)cscx

    ln|cscx-cotx|+c

  • S)sec^2x

    tanx+c

  • S)csc^2x

    -cotx+c

  • S)lnx

    x∙lnx-x+c

  • S)logx

    (x∙lnx-x)/ln10)+c

  • S)logax

    x(logax - logae) + c

  • S)e^x

    e^x+c

  • S)e^k∙x

    1/k ∙ e^k∙x+c

  • S)a^x

    a^x /lna+c

  • S)x^n

    1/(n+1) ∙ x^(n+1)+c

  • S)1/x = x^-1

    ln|x|+c

  • S)√x = x^1/2

    2/3∙(√x)3+c = 2/3 *x^3/2+c

  • S)secx*tanx

    secx

  • S)cscx*cotx

    -cscx

  • S)sech^2x

    tanhx

  • S)csch^2

    -cothx

  • S)sechx*tanhx

    -sechx

  • S)cschx*cothx

    -cschx

  • S)1/√(a^2-x^2)

    sin-1 (x/a)=-cos-1(x/a)

  • S)1/(a^2+x^2)

    1/a tan-1 x/a =-1/a cot x/a

  • S)1/(x√(x^2-a^2))

    1/a sec-1|x|/a=-1/a csc-1|x|/a

  • S)1/√(a^2+x^2)

    sinh-1x/a

  • S)1/√x^2-a^2)

    cosh-1 x/a {a<x}= -cosh-1 -x/a{x<-a}

  • S)1/(a^2-x^2)

    1/a tanh-1x/a {|x|<a} = 1/a coth-1x/a {a^2<x^2}

  • S)1/(x√(a^2-x^2))

    -1/a sech-1 |x|/a {|x|<a}

  • S)1/(x√a^2+x^2)

    -1/a csch-1 |x|/a