Integral of arsinhx using logarithmic form?

I have came across a question which has already took a lot of my time, basically it states that I must use the relationship that arsinhx = ln (x+sqrt (x×x+1))

to integrate arsinhx.

I already know how to integrate arsinhx, but I couldn't really manage to integrate it in logarithmic form.

I tried using by parts and substitutions, but it quickly got messy and very tedious, is there a clean way to achieve this?

Thanks in advance!

3 Answers

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  • cidyah
    Lv 7
    6 years ago
    Favorite Answer

    ∫ arcsinh x dx = ∫ ln ( x+ sqrt(x^2+1) ) dx

    Integrate by parts

    dv= dx; v= x

    u = ln ( x + sqrt(1+x^2) )

    du = 1 / ( x + sqrt(1+x^2) ) d/dx ( x + sqrt(1+x^2) )

    du = 1 / ( x + sqrt(1+x^2) ) ( 1 + 2x / 2sqrt(1+x^2) )

    du = ( 1+ x / sqrt(1+x^2) ) / ( x + sqrt(1+x^2) )

    du = ( sqrt(1+x^2) + x ) / (sqrt(1+x^2) ( x+sqrt(1+x^2))

    du = 1 / sqrt(1+x^2) dx

    ∫ u dv = u v - ∫ v du

    ∫ ln ( x+ sqrt(x^2+1) ) dx = x ln ( x + sqrt(1+x^2) ) - ∫ x dx / sqrt(1+x^2) ----------- (1)

    Consider the integral ∫ x dx / sqrt(1+x^2)

    Let u=1+x^2

    du = 2x dx

    x dx = (1/2) du

    ∫ x dx / sqrt(1+x^2) = (1/2) ∫ du/ sqrt(u) = sqrt(u) = sqrt( 1+x^2)

    substitute this into (1)

    ∫ ln ( x+ sqrt(x^2+1) ) dx = x ln ( x + sqrt(1+x^2) ) - sqrt (1+x^2) + C

  • Ian H
    Lv 7
    6 years ago

    Use relationship that arcsinh(x) = ln[x + √(x^2 + 1)] to integrate arcsinh(x)

    Let 2x = e^y – e^(-y) = 2sinh(y) …………………………(1)

    so that y = arcsinh(x) = sinh^-1(x)

    (e^y)^2 – 2xe^y – 1 = 0

    e^y = x ± √(x^2 + 1), with e^y > 0, so that explains why

    ln(e^y) = y = sinh^-1(x) = ln[x + √(x^2 + 1)] …… (2)

    d/dx[ln(x)*x] = ln(x) + 1, (using the product rule)

    and from that we can work out ∫ln(x)dx = xln(x) – x

    We can use that approach to find the integral of (2) by asuming that

    it will contain the term xln[x + √(x^2 + 1)]

    d/dx{ ln[x + √(x^2 + 1)] * x} = ln[x + √(x^2 + 1)] + T, where

    T = x*d/dx{ ln[x + √(x^2 + 1)]}

    T = x/[x + √(x^2 + 1)] * [1 + x/√(x^2 + 1)] which rearranges as

    T = x/[x + √(x^2 + 1)] * [x + √(x^2 + 1)] /√(x^2 + 1)

    That looks messy until you notice that the squared bracket terms cancel.

    T = x/√(x^2 + 1)] This tells us that

    ∫ln[x + √(x^2 + 1)]dx = xln[x + √(x^2 + 1)] - ∫[x/√(x^2 + 1)]dx

    Substituting back and integrating the last term

    ∫arcsinh(x)dx = x*arcsinh(x) - √(x^2 + 1)

    Regards – Ian H

  • Nick
    Lv 6
    6 years ago

    This will be useful:

    d/dx(e^(arcsinhx)) = e^(arcsinhx)*d/dx(arcsinhx) = 1+x/(x^2+1)^(1/2) = e^(arcsinhx)/(x^2+1)^(1/2)

    => d/dx(arcsinhx) = 1/(x^2+1)^(1/2)

    Now integrate by parts:

    int(arcsinhx dx) = xarcsinhx - int(x/(x^2+1)^(1/2) dx)

    = xarcsinhx - (x^2+1)^(1/2) + constant <----

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