SUMMARY
The integral \(\int \frac{\sqrt{x^4-1}}{x^3}dx\) can be evaluated using trigonometric substitution. The substitution \(x^4 = \sec^2 \theta\) leads to the integral \(\int \frac{\sin^2 \theta}{\cos^{\frac{3}{4}} \theta}d\theta\), which simplifies to \(\int \sec \theta d\theta - \int \cos \theta d\theta\). The final result is \(\ln |x^2+\frac{\sqrt{1-x^2}}{x^2}| - \sqrt{1-x^2} + C\).
PREREQUISITES
- Understanding of integral calculus
- Familiarity with trigonometric identities
- Knowledge of substitution methods in integration
- Ability to manipulate algebraic expressions involving square roots
NEXT STEPS
- Study trigonometric substitution techniques in calculus
- Learn about integration by parts and its applications
- Explore advanced integration techniques such as partial fractions
- Practice solving integrals involving square roots and rational functions
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of trigonometric substitution in action.
