Step-by-Step Guide to Solving Complex Integrals

When facing a complex integral, such as I displaystyle int frac{x^6}{x^2 - 5^4} , dx, a strategic approach is crucial. This article will lead you through the process using integration by parts and partial fractions, providing a detailed, step-by-step guide.

Understanding Integration by Parts

Integration by parts is a powerful technique for solving integrals that are in the form of a product of two functions. The formula is:

[ int u , dv uv - int v , du ]

In the given integral, we start by designating:

[ u x^5 quad text{and} quad dv frac{x}{x^2 - 5^4} , dx ]

Applying integration by parts, we get:

[ I x^5 cdot left(-frac{1}{6x^2 - 5^3}right) - int left(-frac{1}{6x^2 - 5^3}right) cdot 5x^4 , dx ]

This simplifies to:

[ I -frac{x^5}{6x^2 - 5^3} int frac{5x^4}{6x^2 - 5^3} , dx ]

Further Simplification with Integration by Parts

To simplify the integral further, we use integration by parts again with:

[ u frac{5}{6} x^3 quad text{and} quad dv frac{x}{x^2 - 5^3} , dx ]

This yields:

[ I -frac{x^5}{6x^2 - 5^3} cdot frac{5}{6} x^3 cdot left(-frac{1}{4x^2 - 5^2}right) - int frac{-1}{4x^2 - 5^2} cdot frac{5}{2} x^2 , dx ]

Which simplifies to:

[ I -frac{x^5}{6x^2 - 5^3} - frac{5x^3}{24x^2 - 5^2} int frac{5x^2}{8x^2 - 5^2} , dx ]

Final Integration by Parts and Solution Using Partial Fractions

To further simplify, we apply integration by parts one more time with:

[ u frac{5x}{8} quad text{and} quad dv frac{x}{x^2 - 5^2} , dx ]

Resulting in:

[ I -frac{x^5}{6x^2 - 5^3} - frac{5x^3}{24x^2 - 5^2} - frac{5x}{16x^2 - 5} cdot int frac{5}{16x^2 - 5} , dx ]

For the remaining integral, we use partial fractions. The identity for integrals of the form (frac{1}{x^2 - a^2}) is:

[ int frac{1}{x^2 - a^2} , dx frac{1}{2a} ln left| frac{x - a}{x a} right| C ]

Applying this, we get the final result:

[ I boxed{-frac{x^5}{6x^2 - 5^3} - frac{5x^3}{24x^2 - 5^2} - frac{5x}{16x^2 - 5} cdot frac{sqrt{5}}{32} ln left| frac{x - sqrt{5}}{x sqrt{5}} right| C} ]

Key Steps Summary

Identify the structure of the integral and choose appropriate parts for integration by parts. Apply the integration by parts formula multiple times to simplify the expression. Use the identity for integrals involving differences of squares. Combine all parts and constants to form the final answer.

By following these steps, you can tackle even the most complex integrals with confidence. Practice and familiarity with these techniques will greatly enhance your problem-solving skills.