FAC1004 Tutorial 10 — Integration of Hyperbolic & Inverse Hyperbolic Functions
Centre for Foundation Studies in Science
University of Malaya
FAC1004 Advanced Mathematics 2
Topic: Integration of Hyperbolic & Integration Involving Inverse Hyperbolic Functions
Question 1
By using suitable substitution, find the following integrals.
(a) $\int \frac{\sinh\left(\frac{1}{\sqrt{x}}\right)}{(\sqrt{x})^3} dx$
(b) $\int -\cosh x \cdot e^{\ln(\sinh x)^2} dx$
(c) $\int \frac{1}{x\sqrt{(\ln 2x)^2 + 9}} dx$
(d) $\int \frac{\tan^{-1}\left(\frac{x}{2}\right)}{x^2 + 4} dx$
(e) $\int \frac{e^x}{\sqrt{e^{2x} + 1}} dx$
Question 2
Find:
(a) $\int_4^6 \frac{1}{\sqrt{x^2 - 9}} dx$
(b) $\int_3^6 \frac{1}{\sqrt{x^2 + 9}} dx$
(c) $\int_0^{1/2} \frac{2\sin^{-1}(2x)}{\sqrt{1-4x^2}} dx$
(d) $\int_0^{1/3} \frac{3\sinh^{-1}(3x)}{\sqrt{9x^2+1}} dx$
(e) $\int_0^{1/2} \frac{\cosh^{-1}(2x)}{\sqrt{4x^2-1}} dx$
Key Concepts Covered
- Integration by Substitution — u-substitution for hyperbolic integrals
- Definite Integrals — Evaluating with limits
- Inverse Trigonometric Integrals — Forms involving $\sin^{-1}$, $\tan^{-1}$
- Inverse Hyperbolic Integrals — Forms involving $\sinh^{-1}$, $\cosh^{-1}$
- Standard Integral Forms:
- $\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}\left(\frac{x}{a}\right) + C$
- $\int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1}\left(\frac{x}{a}\right) + C$
- Logarithmic Substitution — For integrals involving logarithmic functions