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Glossary

C

Completing the square

Criticality: 3

An algebraic technique used to rewrite a quadratic expression into the form $(x-h)^2+k$ or $a(x-h)^2+k$. This method is crucial for transforming integrands into forms that can be solved using inverse trigonometric functions. [1, 2, 9, 14, 15]

Example:

To integrate ∫13−x2−2xdx\int\frac{1}{\sqrt{3-x^2-2x}}dx, you would use completing the square on the expression under the square root to get 4−(x+1)24-(x+1)^2.

I

Inverse trigonometric functions

Criticality: 3

Functions such as arcsin, arctan, and arcsec, whose derivatives are algebraic expressions. Integrals involving specific quadratic forms often result in these functions, especially after completing the square. [10, 16, 18, 19, 20]

Example:

After completing the square, an integral like ∫1(t−2)2+16dt\int\frac{1}{(t-2)^2+16}dt directly leads to an inverse trigonometric function of the arctan form. [1, 10, 16]

L

Long division

Criticality: 3

An algebraic technique, also known as polynomial long division, used to divide a polynomial by another polynomial. It is particularly useful in integration when the degree of the numerator in a rational function is equal to or greater than the degree of the denominator. [1, 5, 7, 13, 22]

Example:

To simplify 2x2−4x+1\frac{2x^2-4}{x+1} before integrating, you would use long division to rewrite it as 2x−2−2x+12x-2 - \frac{2}{x+1}.

Q

Quadratic expression

Criticality: 2

A polynomial expression of degree 2, typically in the form $ax^2 + bx + c$, where $a \neq 0$. These expressions are often manipulated using completing the square in integration problems. [1, 6, 12, 14]

Example:

The denominator t2−4t+20t^2-4t+20 in the integral ∫4t2−4t+20dt\int\frac{4}{t^2-4t+20}dt is a quadratic expression that can be transformed. [1]

R

Rational function

Criticality: 2

A function that can be expressed as the ratio of two polynomials, where the denominator is not zero. These functions often require algebraic manipulation before integration. [3, 4, 8]

Example:

When integrating ∫x3+2xx2+1dx\int\frac{x^3+2x}{x^2+1}dx, you are dealing with a rational function where the numerator's degree is higher than the denominator's. [22]