FAC1004 Tutorial 10 — Integration of Hyperbolic Functions

Practice problems on integration involving hyperbolic and inverse hyperbolic functions.

Topics Covered

  • Integration using substitution
  • Integrals leading to inverse hyperbolic functions
  • Definite integrals with inverse hyperbolic functions
  • Mixed integration techniques

Problem Set

1. Substitution Method

Find the following integrals using suitable substitution:

  • $\int \frac{\sinh\left(\frac{1}{\sqrt{x}}\right)}{\sqrt{x}} , dx$

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

  • $\int \frac{e^x}{\sqrt{e^{2x} + 1}} , dx$

  • $\int \frac{-\cosh x}{e^x \sqrt{2\tan^{-1}(e^x) - \ln(\sinh x)^2}} , dx$

  • $\int \frac{x^2}{x^2 + 4} , dx$

2. Definite Integrals

Evaluate:

  • $\int_4^6 \frac{dx}{\sqrt{x^2 - 9}}$

  • $\int_3^6 \frac{dx}{\sqrt{x^2 + 9}}$

  • $\int_0^{1/2} \frac{\sin^{-1}(2x)}{\sqrt{1-4x^2}} , dx$

  • $\int_0^{1/3} \frac{\sinh^{-1}(3x)}{\sqrt{9x^2+1}} , dx$

  • $\int_0^{1/2} \frac{\cosh^{-1}(2x)}{\sqrt{4x^2-1}} , dx$

Related

Source File

TUTORIALS_SET_2526/FAC1004 Tutorial 10 25-26.pdf