ENGI 2422 Engineering Mathematics 2

Faculty of Engineering and Applied Science
2008 Winter

Problem Set 3   Questions

Hyperbolic functions, integration by parts, Leibnitz differentiation of an integral.

  1. Find the following derivatives.
    1. d/dx (sech x tanh^2 x)
    2. partial d/dy (sin(2x) cosh^3(5y))     [Hint:   treat x as though it were a constant.]

    3. Prove that the derivative of   csch x   with respect to   x   is   –csch x coth x .

  1. Find the following integrals.
    1. Integ {x^3*e^x + 3*ln(x^2)} dx
    2. Integ {e^(-2x)*cos(3x)} dx
    3. Integ {e^(ax)*cosh(bx)} dx,   where   a and b are any real constants

  1. The function   F (s)   is defined by

    F(s) = Integ_0^oo {e^(-sx)*sin(x) / x} dx

    1. Differentiate   F (s)   with respect to s.
    2. Evaluate your integral expression for dF/ds, using integration by parts.
    3. Hence show that   F (s) = C – Arctan s,   where   C   is an arbitrary constant of integration.
    4. Use an appropriate boundary condition to show that C = pi/2.
    5. Hence evaluate Integ_0^oo {sin x / x} dx exactly.

  1. The suspension cable for a bridge passing through the point   P (x, y) = (0, a) satisfies the equation y = a * cosh(x/a)   (a “catenary” curve).

    1. Determine the element of arc length "ds" for the cable.
    2. Find the total length of the cable from the point at which "x = –5a" to the point at which "x = 5a".
    3. Find the total length of the cable above if the equation defining the shape is replaced by the parabolic approximation y approx= (2.93/a)*x^2 + a instead of the hyperbolic function representation.

  1. Two of the double angle formulae for the circular functions are

    cos 2x   =   cos2x – sin2x
    and
    sin 2x   =   2 sin x cos x .

    Establish the corresponding formulae for the hyperbolic functions   cosh 2x   and   sinh 2x .

  1. The inverse hyperbolic sine function   y = sinh–1x   can be defined by
    x = sinh y = {exp(y) – exp(–y)} / 2

    1. Solve (exp(y) – exp(–y)) / 2 = x for y as a logarithmic function of x.
    2. Differentiate the equation   x = sinh y   [implicitly] with respect to x and use the identity   cosh2x – sinh2x = 1   to express the derivative of   sinh–1x   as a function of x only.
    3. Differentiate your solution to part (a) with respect to x.

    Correct solutions to this question will allow you to conclude that
    Integral { dx / sqrt(1 + x^2) } = arcsinh x = ln(x + sqrt(x^2 + 1))
    Compare this to
    Integral { dx / sqrt(1 – x^2) } = arcsin x

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    Created 2001 01 18 and most recently modified 2007 12 23 by Dr. G.H. George