If the graph of $f(x)$ approaches 0 as $x$ approaches infinity, what does this suggest about the convergence of $\int_a^\infty f(x) dx$ ?

It suggests the integral may converge, but further analysis is needed. The function must approach 0 'fast enough' for convergence.

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If the graph of $f(x)$ approaches 0 as $x$ approaches infinity, what does this suggest about the convergence of $\int_a^\infty f(x) dx$ ?

It suggests the integral may converge, but further analysis is needed. The function must approach 0 'fast enough' for convergence.

If the graph of $f(x)$ oscillates infinitely as $x$ approaches infinity, what does this suggest about the convergence of $\int_a^\infty f(x) dx$ ?

The integral likely diverges, as the area under the curve does not settle to a finite value.

How can you visually identify a discontinuity on a graph that would make an integral improper?

Look for vertical asymptotes or holes within the integration interval.

If the area under the curve of $f(x)$ from $a$ to $b$ is positive, what does this imply about the value of the definite integral $\int_a^b f(x) dx$ ?

The value of the definite integral is positive.

If the area under the curve of $f(x)$ from $a$ to $b$ is negative, what does this imply about the value of the definite integral $\int_a^b f(x) dx$ ?

The value of the definite integral is negative.

How does the shape of the graph of $f(x)$ impact the convergence or divergence of $\int_a^\infty f(x) dx$ ?

If the function decreases rapidly, it is more likely to converge. If it decreases slowly or oscillates, it is more likely to diverge.

What does a graph of $f(x)$ that is always above the x-axis suggest about the integral $\int_a^\infty f(x) dx$ ?

If the integral converges, it will converge to a positive value.

What does a graph of $f(x)$ that is always below the x-axis suggest about the integral $\int_a^\infty f(x) dx$ ?

If the integral converges, it will converge to a negative value.

How can you approximate the value of an improper integral from a graph?

By visually estimating the area under the curve and considering the behavior as x approaches infinity or a point of discontinuity.

If the graph of $f(x)$ has a vertical asymptote at $x=c$ within the interval $[a, b]$ , how does this impact the evaluation of $\int_a^b f(x) dx$ ?

The integral must be split at $x=c$ and evaluated as two separate improper integrals using limits.

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 $e^x$ ?

$\int e^x dx = e^x + C$