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AP Calculus AB 610 Integrating by Polynomial Long Division
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Please subscribe! / nickperich • AP Calculus AB 6.10 Integrating by Polynomial Long Division • *Overview:* • This section focuses on the technique of integrating rational functions by using polynomial long division. This method is particularly useful when the degree of the numerator is greater than or equal to the degree of the denominator, allowing for simplification before integration. • *Key Concepts:* • 1. *Polynomial Long Division:* • When you have a rational function in the form \\( \\frac{P(x)}{Q(x)} \\), where \\( P(x) \\) is a polynomial of equal or higher degree than \\( Q(x) \\), polynomial long division can be applied to rewrite the rational function in a more manageable form. • The result of the long division will be a polynomial plus a remainder term, which can be expressed as: • \\[ • \\frac{P(x)}{Q(x)} = D(x) + \\frac{R(x)}{Q(x)} • \\] • where \\( D(x) \\) is the quotient polynomial and \\( R(x) \\) is the remainder polynomial with a degree less than that of \\( Q(x) \\). • 2. *Example Problem:* • Consider the integral: • \\[ • \\int \\frac{2x^2 + 3x + 1}{x + 1} \\, dx. • \\] • Since the degree of the numerator (2) is greater than the degree of the denominator (1), we perform polynomial long division: • Divide \\( 2x^2 \\) by \\( x \\) to get \\( 2x \\). • Multiply \\( 2x \\) by \\( (x + 1) \\) to get \\( 2x^2 + 2x \\). • Subtract \\( (2x^2 + 2x) \\) from \\( (2x^2 + 3x + 1) \\) to get \\( x + 1 \\). • Now divide \\( x + 1 \\) by \\( x + 1 \\) to get \\( 1 \\) with a remainder of \\( 0 \\). • Thus, the result of the division is: • \\[ • 2x + 1. • \\] • The integral now simplifies to: • \\[ • \\int (2x + 1) \\, dx. • \\] • 3. *Integration of the Result:* • Now integrate term by term: • \\[ • \\int (2x + 1) \\, dx = x^2 + x + C, • \\] • where \\( C \\) is the constant of integration. • 4. *Combining Results:* • After performing polynomial long division and integrating, always remember to express the final answer in a clear format. In this example, we have: • \\[ • \\int \\frac{2x^2 + 3x + 1}{x + 1} \\, dx = x^2 + x + C. • \\] • 5. *Application:* • This technique is useful for more complex integrals involving rational functions and can be applied to evaluate definite integrals as well. Students should practice identifying when polynomial long division is necessary and work through various examples to gain proficiency. • Conclusion • Integrating by polynomial long division is an essential skill in AP Calculus for handling rational functions. By transforming a complex integral into a simpler form through division, students can effectively integrate and obtain meaningful results. Mastery of this technique not only enhances problem-solving skills but also deepens understanding of polynomial behavior and its applications in calculus. • I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: • / nickperich • Nick Perich • Norristown Area High School • Norristown Area School District • Norristown, Pa • #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study
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