How do you express an improper integral with an upper bound of infinity as a limit?
$\int_a^\infty f(x) , dx = \lim_{b \to \infty} \int_a^b f(x) , dx$
How do you express an improper integral with a lower bound of negative infinity as a limit?
$\int_{-\infty}^b f(x) , dx = \lim_{a \to -\infty} \int_a^b f(x) , dx$
How do you express an improper integral with both bounds being infinity as a limit?
$\int_{-\infty}^\infty f(x) , dx = \lim_{a \to -\infty} \int_a^c f(x) , dx + \lim_{b \to \infty} \int_c^b f(x) , dx$
What is the formula for the integral of $\frac{1}{x}$?
$\int \frac{1}{x} dx = \ln|x| + C$
What is the formula for the integral of $\frac{1}{a^2+x^2}$?
$\int \frac{1}{a^2+x^2} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C$
Give the general form of partial fraction decomposition.
$\frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{x-b} + ...$
What is the formula for integration by substitution?
$\int f(g(x))g'(x)dx = \int f(u)du$ where $u=g(x)$
What is the formula for the integral of $e^x$?
$\int e^x dx = e^x + C$
What is the formula for the integral of $x^n$?
$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$
What is the formula for the integral of $sin(x)$?
$\int sin(x) dx = -cos(x) + C$
What does the First Fundamental Theorem of Calculus state?
If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$.
How does the Squeeze Theorem relate to improper integrals?
If $g(x) \leq f(x) \leq h(x)$ and $\int_a^\infty g(x) dx$ and $\int_a^\infty h(x) dx$ both converge to the same limit, then $\int_a^\infty f(x) dx$ also converges to that limit.
What is the Comparison Theorem for Improper Integrals?
If $0 \leq f(x) \leq g(x)$ for all $x \geq a$, then if $\int_a^\infty g(x) dx$ converges, so does $\int_a^\infty f(x) dx$, and if $\int_a^\infty f(x) dx$ diverges, so does $\int_a^\infty g(x) dx$.
What is the Second Fundamental Theorem of Calculus?
$\int_a^b F'(x) dx = F(b) - F(a)$
How does the Limit Comparison Test relate to improper integrals?
If $\lim_{x \to \infty} \frac{f(x)}{g(x)} = c$, where $0 < c < \infty$, then $\int_a^\infty f(x) dx$ and $\int_a^\infty g(x) dx$ either both converge or both diverge.
What is the theorem for integration by parts?
$\int u dv = uv - \int v du$
What is the theorem for u-substitution?
$\int f(g(x))g'(x)dx = \int f(u)du$ where $u=g(x)$
What is the theorem for partial fraction decomposition?
Decompose a rational function into simpler fractions that are easier to integrate.
What is the theorem for the integral of $\frac{1}{x}$?
$\int \frac{1}{x} dx = \ln|x| + C$
What is the theorem for the integral of $e^x$?
$\int e^x dx = e^x + C$
What is the difference between evaluating $\int_a^b f(x) dx$ and $\int_a^\infty f(x) dx$?
Definite Integral: Direct evaluation using the Fundamental Theorem of Calculus. Improper Integral: Requires expressing as a limit and evaluating the limit.
Compare and contrast the convergence tests for series and improper integrals.
Series: Ratio test, comparison test, etc. Improper Integrals: Direct evaluation via limits, comparison theorems (comparing to known convergent/divergent integrals).
What is the difference between a definite integral and an indefinite integral?
Definite Integral: Has upper and lower bounds, evaluates to a numerical value. Indefinite Integral: Does not have bounds, evaluates to a function + C.
Compare the methods for handling discontinuities within the interval of integration for proper and improper integrals.
Proper Integrals: Discontinuities typically lead to undefined integrals. Improper Integrals: Discontinuities are handled by splitting the integral and using limits.
What is the difference between convergence and divergence?
Convergence: The limit approaches a finite value. Divergence: The limit approaches infinity or does not exist.
Compare the use of u-substitution in proper and improper integrals.
Proper Integrals: u-substitution simplifies the integrand. Improper Integrals: u-substitution simplifies the integrand before taking the limit.
Compare the use of integration by parts in proper and improper integrals.
Proper Integrals: Integration by parts helps to solve the integral. Improper Integrals: Integration by parts helps to solve the integral before taking the limit.
Compare the use of partial fraction decomposition in proper and improper integrals.
Proper Integrals: Partial fraction decomposition helps to solve the integral. Improper Integrals: Partial fraction decomposition helps to solve the integral before taking the limit.
Compare the use of the Fundamental Theorem of Calculus in proper and improper integrals.
Proper Integrals: Fundamental Theorem of Calculus is used to solve the integral. Improper Integrals: Fundamental Theorem of Calculus is used to solve the integral before taking the limit.
What is the difference between the integral of $\frac{1}{x}$ and $\frac{1}{x^2}$?
$\int \frac{1}{x} dx = ln|x| + C$ and $\int \frac{1}{x^2} dx = -\frac{1}{x} + C$
