Calculus 2 2026 05 27

Vocabulary and key concepts from a lecture covering techniques of integration and improper integrals.

Improper Integral

An integral where the bounds are infinite or there is a discontinuity in the upper bound, lower bound, or somewhere in between.

Convergence

The property of an improper integral when it results in a finite value rather than being undefined or going to infinity.

Discontinuity in Integrals

A point where the denominator of the function is equal to zero, requiring the integral to be split into two parts.

Integration by Parts

A technique used for integration, often applied to products like $2x \times e^x$, based on the selection of $u$ and $dv$.

Trigonometric Substitution

A method used to solve integrals involving expressions like $\text{sqrt}(a^2 - x^2)$ by using triangle relationships and trigonometric identities.

Partial Fraction Decomposition

A technique for integrating rational functions by breaking them down into a sum of simpler fractions based on the factors of the denominator.

Rational Function

A function that is a ratio of two polynomials, indexed by the degrees of its numerator and denominator.

Indeterminate Form

Mathematical expressions such as $0 \times \text{infinity}$ or $\frac{\text{infinity}}{\text{infinity}}$ that require specific limit evaluation techniques.

Exponents over Roots

The rule for rewriting a radical as a power, where the exponent becomes the numerator and the root becomes the denominator, such as the seventh root of $x^8$ being $x^{8/7}$.

Arc Tan Horizontal Asymptotes

The values that the limit of $\text{arctan}(x)$ approaches as $x$ goes to infinity ($\frac{\text{pi}}{2}$) or negative infinity ($-\frac{\text{pi}}{2}$).

Double Angle Formula for Sine

The trigonometric identity $\text{sin}(2\theta) = 2 \text{sin}(\theta)\text{cos}(\theta)$.

Half-Angle Identity for $\text{sin}^2(3x)$

The identity used to reduce powers for integration: $\frac{1}{2} - \frac{1}{2}\text{cos}(6x)$, where the angle is doubled.

Odd Power Strategy

In integrals with powers of sine and cosine, the term with the odd power is typically preserved while the other is used for u-substitution.