Evaluate the integral. ∫(x^3+3 x^2+x+9)/((x^2+1)(x^2+3)) d x...

Name: Problem 31, Chapter 7: Calculus Early Transcendentals
Uploaded: 2025-07-21T20:46:10-08:00
Duration: 2 min 29 s
Channel: Nick Johnson
Description: Evaluate the integral. ∫(x^3+3 x^2+x+9)/((x^2+1)(x^2+3)) d x

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Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In integration, these functions often require special techniques because their antiderivatives are not always obvious without manipulation, such as dividing polynomials or decomposing the expression.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to rewrite a rational function as a sum of simpler fractions whose denominators are polynomials of lower degree. This method simplifies the integration process by reducing the original fraction into a series of easier-to-integrate components.

Integration of Rational Functions

Integrating rational functions usually involves first expressing the integrand in a simpler form through algebraic techniques like polynomial division or partial fraction decomposition. Once decomposed, standard integration techniques and formulas can be applied to each term individually.

Standard Integral Forms

Standard integral forms, such as the integral of 1/(x^2 + a^2) leading to an arctan function or logarithmic forms from linear denominators, are essential tools in calculus. Familiarity with these forms allows for the quick evaluation of many integrals once a rational function has been appropriately decomposed.

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