Calculus II: Power Reduction and Integration by Parts Flashcards

This flashcard set covers power reduction formulas for trigonometric functions and various strategies for Integration by Parts (IBP) as discussed in the lecture.

Secant Power Reduction Formula

An algorithmic method used to solve powers of secant by reducing the power by two: $\int \sec^n(x)\,dx = \frac{1}{n-1}\sec^{n-2}(x)\tan(x) + \frac{n-2}{n-1} \int \sec^{n-2}(x)\,dx$.

Tangent Power Reduction Formula

A formula that reduces the power of a tangent integral by two each step: $\int \tan^n(x)\,dx = \frac{1}{n-1}\tan^{n-1}(x) - \int \tan^{n-2}(x)\,dx$.

Integration by Parts

A technique for integration based on the product rule, represented by the formula $\int u\,dv = uv - \int v\,du$.

Selection of $u$ in Integration by Parts

The part of the integral that should be chosen so it becomes simpler or reduced in complexity when its derivative ($du$) is taken.

Selection of $dv$ in Integration by Parts

The part of the integral chosen to be easily anti-differentiated to find $v$.

Chain Rule (in du calculation)

The rule applied when taking the derivative of a composite function, such as $\ln(x)^4$, resulting in $4\ln(x)^3 \times \frac{1}{x}$.

Cyclic Integration by Parts

A scenario occurring with functions like $e^x \sin(x)$ where the original integral reappears after applying integration by parts twice, allowing it to be solved algebraically.

Definite Integration by Parts

The application of bounds to the integration by parts formula: $[uv]_a^b - \int_a^b v\,du$.

Derivative of Inverse Tangent ($\arctan(x)$)

The derivative used in IBP when $u = \tan^{-1}(x)$, which is $\frac{1}{x^2+1}$.

Substitution of the Denominator

A technique used to simplify integrals like $\int \frac{x}{x^2+1}\,dx$ by letting $u = x^2+1$ and setting $du = 2x\,dx$.

Algebraic Simplification Trick

A method used when the numerator and denominator have similar degree, allowing for a variable substitution (like $w = 3r - 4$) to split the fraction.