Chapter 8: Integration Techniques

Flashcards for reviewing integration techniques, trigonometric integrals and substitution, partial fractions, numerical integration, and improper integrals.

Integration Techniques

Methods to evaluate integrals, including rewriting the integrand using trig identities, splitting up fractions, and completing the square.

Multiplication by 1

A strategy to rewrite the integrand to evaluate integrals.

Trig Identities

Using trigonometric identities to rewrite the integrand.

Splitting Up Fractions

A technique to simplify complex rational functions for integration.

Completing the Square

Algebraic manipulation to transform a quadratic expression into a more integrable form.

Integration by Parts

A technique to integrate the product of two functions using the formula ∫u dv = uv - ∫v du.

Product Rule for Differentiation

d/dx [u*v] = u dv + v du

LIATE Rule

A guideline for choosing 'u' in integration by parts: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.

Right Triangle Trigonometry

sin(θ) = Opposite/Hypotenuse, cos(θ) = Adjacent/Hypotenuse, tan(θ) = Opposite/Adjacent

Trigonometric Integrals

Integrals involving trigonometric functions, often solved by using trigonometric identities.

Trigonometric Identities

Equations involving trigonometric functions that are true for all values of the variables.

Trigonometric Substitution

A technique to simplify integrals containing square roots by substituting trigonometric functions.

Partial Fractions

A method to decompose a rational function into simpler fractions that are easier to integrate.

Integration Strategies

Various techniques to evaluate integrals, including substitution, integration by parts, trigonometric integrals, trigonometric substitution, partial fractions, etc.

Numerical Integration

Approximating the value of a definite integral using numerical methods (Midpoint Rule, Trapezoid Rule, Simpson’s Rule).

Midpoint Rule

Approximating a definite integral using rectangles whose heights are determined by the function value at the midpoint of each subinterval.

Trapezoid Rule

Approximating a definite integral using trapezoids to estimate the area under a curve.

Simpson's Rule

Approximating a definite integral using parabolas to estimate the area under a curve.

Absolute Error

The difference between an approximate value and the exact value: |Q - x|.

Relative Error

The absolute error divided by the exact value: |(Q - x) / x|.

Improper Integrals

Integrals where one or both limits of integration are infinite or the integrand has a vertical asymptote within the interval of integration.

Converging Improper Integral

An improper integral that has a finite value.

Diverging Improper Integral

An improper integral that does not have a finite value (i.e., it goes to infinity or does not exist).