FAC1004 Tutorial 9 — Inverse Hyperbolic Functions

Practice problems on derivatives of inverse hyperbolic functions and integration techniques.

Topics Covered

• Derivatives of inverse hyperbolic functions
• Integration leading to inverse hyperbolic functions
• Logarithmic forms of integrals
• Substitution techniques

Problem Set

1. Derivatives of Inverse Hyperbolic: Find $\frac{dy}{dx}$ for:

   • $y = \cosh^{-1}(5x - 7)$
   • $y = \text{sech}^{-1}(\ln x)$
   • $y = \ln(\tanh^{-1} x)$
   • $y = \sinh^{-1}(x^{-3})$

2. Advanced Differentiation: Differentiate:

   • $y = x^2\cosh^{-1}(6x^2 - 7x^{-2})$
   • $y = \cos(\sinh^{-1}(x^6))$
   • $y = \frac{\sinh^{-1}(2x^2)}{\tanh^{-1}(4x + x^{-2})}$

3. Prove Integration Formulas: Show that:

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

4. Evaluate Integrals:

   • $\int \sinh^6 x \cosh(x) , dx$
   • $\int \cosh(2x - 3) , dx$
   • $\int \sqrt{\tanh x}\text{ sech}^2 x , dx$
   • $\int \frac{dx}{\sqrt{1+9x^2}}$
   • $\int \frac{dx}{\sqrt{9x^2 - 25}}$
   • $\int \frac{dx}{x\sqrt{1+4x^2}}$

Related

• FAC1004 - Advanced Mathematics II (Computing) — main course page
• Hyperbolic Functions — concept page
• FAC1004 L19-L20 — Inverse Hyperbolic Functions — related lecture
• FAC1004 L21-L22 — Integrals Involving Hyperbolic Functions — related lecture

Source File

TUTORIALS_SET_2526/FAC1004 Tutorial 9 25-26.pdf