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Mastering integration through polynomial long division is essential in calculus for handling rational functions with numerators of higher or equal degree compared to the denominators. This method simplifies the integration process by breaking down the original rational function into a sum of a polynomial and a proper fraction, each solvable with standard integration techniques. Practical examples demonstrate the effectiveness of long division in making complex integrals more manageable.
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Polynomial long division is a fundamental technique in calculus for integrating rational functions
Simplification
This method streamlines the integration process by dividing the higher-degree polynomial numerator by the lower-degree polynomial denominator
Decomposition
The integral of the original rational function is decomposed into the sum of the integral of a polynomial and the integral of a proper rational function
Each of the components can be tackled using standard integration methods
The use of long division in integration is appropriate when the integrand is a rational function where the polynomial in the numerator has a degree that is greater than or equal to the degree of the polynomial in the denominator
When the numerator's degree is less, alternative methods such as integration by parts or partial fraction decomposition are more suitable
The degree of a polynomial is the highest power of its variable
To perform integration using long division, one must first identify the polynomials within the integrand
The next step is to divide the polynomial in the numerator by the polynomial in the denominator using polynomial long division
This division simplifies the original integral into components that are more amenable to integration using established techniques
Polynomial long division operates on the same principles as long division with numbers
The process involves repeatedly subtracting multiples of the divisor polynomial from the dividend polynomial to find the quotient and remainder
The remainder becomes part of the new, simplified integrand that is to be integrated
00
The process simplifies integration by dividing a higher-degree ______ numerator by a lower-degree ______ denominator, resulting in an expression easier to ______.
polynomial
polynomial
integrate
01
Long division in integration: numerator's degree vs. denominator's.
Use when numerator's degree >= denominator's degree.
02
Alternative methods when numerator's degree < denominator's degree.
Use integration by parts or partial fraction decomposition.
