calculus

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using tangent half substitution show that integral of
sec(x)=ln(secx+tanx)

  • calculus -

    If tan(x/2)=u then
    dx=2du/(1+u^2), sec(x)=(1+u^2)/(1-u^2)

    Integral of sec(x)dx=Integral of 2du/(1-u^2)=ln((1+u)/(1-u))

    (1+tan(x/2))/(1-tan(x/2))=
    (cos(x/2)+sin(x/2))/(cos(x/2)-sin(x/2))=

    Multiply numerator and denominator by
    cos(x/2)+sin(x/2)

    =(1+sin(x))/cos(x)=sec(x)+tan(x)

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