Explain when to use long division when integrating rational functions.
Use long division when the degree of the numerator is greater than or equal to the degree of the denominator.
Explain the purpose of completing the square in integration.
Completing the square transforms a quadratic expression into a form that can be easily integrated, often leading to inverse trigonometric functions.
Why do we manipulate the numerator to 1 when completing the square?
Making the numerator 1 simplifies the integral and makes it easier to recognize inverse trigonometric integral forms.
Steps for integrating \(\int \frac{P(x)}{Q(x)} dx\) where deg(P) ≥ deg(Q).
1. Perform polynomial long division to get \(P(x)/Q(x) = S(x) + R(x)/Q(x)\). 2. Integrate \(\int S(x) dx\) and \(\int \frac{R(x)}{Q(x)} dx\) separately. 3. Combine the results.
How to integrate \(\int \frac{1}{ax^2 + bx + c} dx\)?
1. Complete the square in the denominator. 2. Use a u-substitution to simplify. 3. Apply the appropriate inverse trigonometric integral formula.
How do you evaluate \(\int \frac{2x^2-4}{x+1}dx\)?
1. Use polynomial long division: \(\frac{2x^2-4}{x+1} = 2x - 2 - \frac{2}{x+1}\). 2. Integrate: \(\int (2x - 2 - \frac{2}{x+1}) dx = x^2 - 2x - 2\ln|x+1| + C\).
How do you evaluate \(\int \frac{4}{t^2-4t+20}dt\)?
1. Complete the square: \(t^2 - 4t + 20 = (t-2)^2 + 16\). 2. Rewrite the integral: \(\int \frac{4}{(t-2)^2 + 16} dt\). 3. Use arctan integral form: \(\arctan(\frac{t-2}{4}) + C\).
How do you evaluate \(\int \frac{2x^3+3x^2-17x-27}{x^2-9}dx\)?
1. Use polynomial long division: \(\frac{2x^3+3x^2-17x-27}{x^2-9} = 2x+3 + \frac{x}{x^2-9}\). 2. Integrate: \(\int (2x+3 + \frac{x}{x^2-9}) dx = x^2+3x+\frac{1}{2}ln(|x^2-9|) + C\).
How do you evaluate \(\int \frac{1}{\sqrt{3-x^2-2x}}dx\)?
1. Complete the square: \(-x^2-2x+3 = -(x+1)^2 + 4\). 2. Rewrite the integral: \(\int \frac{1}{\sqrt{-(x+1)^2 + 4}} dx\). 3. Use arcsin integral form: \(\arcsin(\frac{1}{2}(x+1)) + C\).
How do you evaluate \(\int \frac{x^2}{x+1}dx\)?
1. Use polynomial long division: \(\frac{x^2}{x+1} = x-1 + \frac{1}{x+1}\). 2. Integrate: \(\int (x-1 + \frac{1}{x+1}) dx = \frac{1}{2}x^2-x+ln(|x+1|) + C\).
Define a rational function.
A function that can be expressed as the quotient of two polynomials.
What is completing the square?
A technique to rewrite a quadratic expression in the form $(x-h)^2 + k$.
Define an integrand.
The function that is being integrated.
What is polynomial long division?
An algorithm for dividing a polynomial by another polynomial of the same or lower degree.
