FAC1004 Tutorial 9 — Inverse Hyperbolic Functions

Centre for Foundation Studies in Science
University of Malaya
FAC1004 Advanced Mathematics 2

Topic: Inverse Hyperbolic Functions

Question 1

Find $\frac{dy}{dx}$ if:

(a) $y = \cosh^{-1}(5x - 7)$

(b) $y = \text{sech}^{-1}(\ln x)$

(c) $y = \ln(\tanh x)$

(d) $y = \sinh^{-1}(x^{-3})$

Question 2

Differentiate the following with respect to $x$.

(a) $y = x^2 \cosh^{-1}(6x^2 - 7x^{-2})$

(b) $y = \cos(\sinh^{-1} x^6)$

(c) $y = \frac{\sinh^{-1}(2x)}{(\tanh^{-1}(4x + x^{-2}))^{-4}}$

Question 3

Show that:

(a) $\int \frac{dx}{\sqrt{a^2 + x^2}} = \ln(x + \sqrt{x^2 + a^2}) + C$

(b) $\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln(x + \sqrt{x^2 - a^2}) + C$

(c) $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$

Question 4

Evaluate the following integrals.

(a) $\int \sinh^6 x \cosh(x) dx$

(b) $\int \cosh(2x - 3) dx$

(c) $\int \sqrt{\tanh x} \text{ sech}^2 x dx$

(d) $\int \frac{dx}{\sqrt{1 + 9x^2}}$

(e) $\int \frac{dx}{\sqrt{9x^2 - 25}}$

(f) $\int \frac{dx}{x\sqrt{1 + 4x^2}}$

Key Concepts Covered

  • Inverse Hyperbolic Functions — Derivative formulas
  • $\frac{d}{dx}[\sinh^{-1} x] = \frac{1}{\sqrt{x^2+1}}$
  • $\frac{d}{dx}[\cosh^{-1} x] = \frac{1}{\sqrt{x^2-1}}$
  • $\frac{d}{dx}[\tanh^{-1} x] = \frac{1}{1-x^2}$
  • Integration by Substitution — u-substitution techniques
  • Hyperbolic Substitution — For integrals involving $\sqrt{a^2+x^2}$, $\sqrt{x^2-a^2}$
  • Chain Rule Applications — Differentiating composite inverse hyperbolic functions

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