Chapter 1 Flashcards
What is the formula for integration by substitution?
If u = g(x), then ∫ f(g(x)g’(x))dx = ∫ f(u)du
This technique simplifies integration by changing variables.
What is the first step in the integration by substitution process?
Determine u = g(x)
This involves identifying a suitable function to substitute.
How do you express dx in terms of du during substitution?
dx = (1/g’(x))du
This is derived from the relationship between u and x.
What is the result of the integral ∫(x² + 2x + 3)6(x + 1)dx after substitution?
1/14(x² + 2x + 3)⁷ + C
C is the constant of integration.
In the example ∫x⁻²(1 + 2/x)⁵dx, what substitution is made?
Let u = 1/x + 2
This simplifies the integral for easier evaluation.
What is the integral of sin²(x)e^(cos²(x))dx using substitution?
-1/2 e^(cos²(x)) + C
The solution involves integrating with respect to u.
What is the integral of 1/u du?
ln|u| + C
This is a standard result in integration involving logarithmic functions.
What is the integral of e^u du?
e^u + C
This is a basic property of exponential functions.
What is the rule for integration by parts?
∫u dv = uv - ∫v du
This rule corresponds to the product rule of differentiation.
What is tabular integration?
A method used for repeated integration by parts
It simplifies the process when many repetitions are needed.
What is the definition of improper integrals?
Integrals where the interval of integration is infinite or includes a singularity
This includes Type I and Type II improper integrals.
What are the two types of improper integrals?
- Type I: Infinite Limits of Integration
- Type II: Discontinuous Integrand
These types help classify the behavior of the integral.
What does it mean for an improper integral to converge?
The limit used to define the integral exists and is finite
Conversely, if the limit does not exist, it diverges.
What is the improper integral ∫[1, ∞] (1/x^p)dx converges for what values of p?
p > 1
This determines the conditions under which the integral converges.
In integration involving trigonometric functions, what is a common substitution for sin(x)?
u = cos(x)
This substitution simplifies integrals involving sine and cosine.
What is the integral of sec²(x)tan(x)dx?
sec²(x) + C
This is a standard integral in calculus involving trigonometric functions.
What is the result of the integral ∫tan(x)dx?
-ln|cos(x)| + C
This can also be expressed as ln|sec(x)| + C.
What is the method for evaluating improper integrals?
Calculate the limit of the integral as it approaches the point of discontinuity or infinity
This is essential for determining convergence.
What is the formula for integration by substitution?
∫ f(g(x))g’(x)dx = ∫ f(u)du
Ehherh
((x^2 + 2x + 3)^7)/7 + C
What is the method of Integration by Substitution?
Integration by Substitution involves letting u = g(x) so that du = g′(x) dx, which transforms an integral ∫ f(g(x)) g′(x) dx into ∫ f(u) du.
What is the formula for Integration by Parts?
Integration by Parts is given by ∫ u dv = uv − ∫ v du, derived from the product rule for differentiation.
How are rational functions integrated using Partial Fractions?
The method involves decomposing a rational function into a sum of simpler fractions (with linear or quadratic denominators) that can be integrated individually.
What are Trigonometric Techniques of Integration?
These techniques use trigonometric identities and substitutions to simplify integrals involving trigonometric functions into more manageable forms.
What are Improper Integrals and how are they evaluated?
Improper Integrals involve infinite limits or discontinuous integrands and are evaluated by replacing the problematic limit with a variable and taking its limit (e.g. ∫ₐ∞ f(x) dx = lim₍b→∞₎ ∫ₐᵇ f(x) dx).
Solve ∫ dx/(x² + 2x + 3) using substitution.
Complete the square: x² + 2x + 3 = (x+1)² + 2. Let u = x + 1 so that the integral becomes ∫ du/(u² + 2), which evaluates to (1/√2) arctan(u/√2) + C, i.e. (1/√2) arctan((x+1)/√2) + C.
Evaluate ∫ (5x − 23)/(6x² − 11x − 7) dx using partial fractions.
Factor the denominator as (2x + 1)(3x − 7). Write (5x − 23)/(6x² − 11x − 7) = A/(2x + 1) + B/(3x − 7). Solving gives A = 3 and B = −2. Thus, the integral is (3/2) ln|2x + 1| − (2/3) ln|3x − 7| + C.
Evaluate the definite integral ∫₋π/₃⁰ cos x · sin³ x dx.
Let u = sin x (so du = cos x dx). Changing limits: when x = −π/3, u = −√3/2; when x = 0, u = 0. The integral becomes ∫₋√3/₂⁰ u³ du = [u⁴/4]₋√3/₂⁰ = −(9/64), so the value is −9/64.
For which values of p does the improper integral ∫₁∞ 1/xᵖ dx converge?
It converges if and only if p > 1; for p ≤ 1 the integral diverges.
Does the improper integral ∫₁∞ 1/x dx converge?
No – since p = 1 the integral diverges (it grows logarithmically without bound).
Is the area under y = (ln x)/x² from x = 1 to ∞ finite, and if so, what is it?
Yes, it is finite. Using integration by parts one finds ∫₁∞ (ln x)/x² dx = 1.
What is Tabular Integration and when is it used?
Tabular Integration is a streamlined method for performing repeated integration by parts, especially when one function differentiates repeatedly to zero. It organizes derivatives and integrals in a table to simplify the process.
