How do you evaluate $\int_0^\infty e^{-x} dx$?
1) Express as a limit: $\lim_{b \to \infty} \int_0^b e^{-x} dx$. 2) Evaluate the integral: $\lim_{b \to \infty} [-e^{-x}]_0^b = \lim_{b \to \infty} (-e^{-b} + e^0)$. 3) Evaluate the limit: $\lim_{b \to \infty} (-e^{-b} + 1) = 1$. 4) The integral converges to 1.
How do you evaluate $\int_1^\infty \frac{1}{x^2} dx$?
1) Express as a limit: $\lim_{b \to \infty} \int_1^b \frac{1}{x^2} dx$. 2) Evaluate the integral: $\lim_{b \to \infty} [-\frac{1}{x}]_1^b = \lim_{b \to \infty} (-\frac{1}{b} + 1)$. 3) Evaluate the limit: $\lim_{b \to \infty} (-\frac{1}{b} + 1) = 1$. 4) The integral converges to 1.
How do you evaluate $\int_0^1 \frac{1}{\sqrt{x}} dx$?
1) Express as a limit: $\lim_{a \to 0^+} \int_a^1 \frac{1}{\sqrt{x}} dx$. 2) Evaluate the integral: $\lim_{a \to 0^+} [2\sqrt{x}]_a^1 = \lim_{a \to 0^+} (2 - 2\sqrt{a})$. 3) Evaluate the limit: $\lim_{a \to 0^+} (2 - 2\sqrt{a}) = 2$. 4) The integral converges to 2.
How do you evaluate $\int_1^\infty \frac{1}{x} dx$?
1) Express as a limit: $\lim_{b \to \infty} \int_1^b \frac{1}{x} dx$. 2) Evaluate the integral: $\lim_{b \to \infty} [\ln|x|]_1^b = \lim_{b \to \infty} (\ln(b) - \ln(1))$. 3) Evaluate the limit: $\lim_{b \to \infty} (\ln(b) - 0) = \infty$. 4) The integral diverges.
How do you evaluate $\int_{-\infty}^0 xe^x dx$?
1) Express as a limit: $\lim_{a \to -\infty} \int_a^0 xe^x dx$. 2) Integrate by parts: $u=x, dv=e^x dx$, so $du=dx, v=e^x$. Then $\int xe^x dx = xe^x - \int e^x dx = xe^x - e^x + C$. 3) Evaluate the integral: $\lim_{a \to -\infty} [xe^x - e^x]_a^0 = \lim_{a \to -\infty} [(0 - e^0) - (ae^a - e^a)] = \lim_{a \to -\infty} [-1 - ae^a + e^a]$. 4) Evaluate the limit: $\lim_{a \to -\infty} [-1 - ae^a + e^a] = -1 - 0 + 0 = -1$. 5) The integral converges to -1.
How do you evaluate $\int_{-\infty}^{\infty} \frac{1}{1+x^2} dx$?
1) Split the integral: $\int_{-\infty}^{\infty} \frac{1}{1+x^2} dx = \int_{-\infty}^{0} \frac{1}{1+x^2} dx + \int_{0}^{\infty} \frac{1}{1+x^2} dx$. 2) Express as limits: $\lim_{a \to -\infty} \int_a^0 \frac{1}{1+x^2} dx + \lim_{b \to \infty} \int_0^b \frac{1}{1+x^2} dx$. 3) Evaluate the integral: $\lim_{a \to -\infty} [\arctan(x)]_a^0 + \lim_{b \to \infty} [\arctan(x)]_0^b = \lim_{a \to -\infty} [\arctan(0) - \arctan(a)] + \lim_{b \to \infty} [\arctan(b) - \arctan(0)]$. 4) Evaluate the limits: $[0 - (-\frac{\pi}{2})] + [\frac{\pi}{2} - 0] = \frac{\pi}{2} + \frac{\pi}{2} = \pi$. 5) The integral converges to $\pi$.
How do you evaluate $\int_2^{\infty} \frac{1}{x(x-1)} dx$?
1) Express as a limit: $\lim_{b \to \infty} \int_2^b \frac{1}{x(x-1)} dx$. 2) Partial fraction decomposition: $\frac{1}{x(x-1)} = \frac{A}{x} + \frac{B}{x-1}$. Solving gives $A = -1$ and $B = 1$. So, $\frac{1}{x(x-1)} = \frac{-1}{x} + \frac{1}{x-1}$. 3) Evaluate the integral: $\lim_{b \to \infty} \int_2^b (\frac{-1}{x} + \frac{1}{x-1}) dx = \lim_{b \to \infty} [-\ln|x| + \ln|x-1|]_2^b = \lim_{b \to \infty} [\ln|\frac{x-1}{x}|]_2^b = \lim_{b \to \infty} [\ln|\frac{b-1}{b}| - \ln|\frac{2-1}{2}|]$. 4) Evaluate the limit: $\lim_{b \to \infty} [\ln|\frac{b-1}{b}| - \ln(\frac{1}{2})] = \ln(1) - \ln(\frac{1}{2}) = 0 - (-\ln(2)) = \ln(2)$. 5) The integral converges to $\ln(2)$.
How do you evaluate $\int_0^{\infty} cos(x) dx$?
1) Express as a limit: $\lim_{b \to \infty} \int_0^b cos(x) dx$. 2) Evaluate the integral: $\lim_{b \to \infty} [sin(x)]_0^b = \lim_{b \to \infty} (sin(b) - sin(0))$. 3) Evaluate the limit: $\lim_{b \to \infty} (sin(b) - 0)$. Since $sin(b)$ oscillates between -1 and 1 as b approaches infinity, the limit does not exist. 4) The integral diverges.
How do you evaluate $\int_0^{3} \frac{1}{x-2} dx$?
1) Split the integral at the discontinuity: $\int_0^{3} \frac{1}{x-2} dx = \int_0^{2} \frac{1}{x-2} dx + \int_2^{3} \frac{1}{x-2} dx$. 2) Express as limits: $\lim_{b \to 2^-} \int_0^b \frac{1}{x-2} dx + \lim_{a \to 2^+} \int_a^3 \frac{1}{x-2} dx$. 3) Evaluate the integral: $\lim_{b \to 2^-} [\ln|x-2|]_0^b + \lim_{a \to 2^+} [\ln|x-2|]_a^3 = \lim_{b \to 2^-} [\ln|b-2| - \ln|-2|] + \lim_{a \to 2^+} [\ln|3-2| - \ln|a-2|] = \lim_{b \to 2^-} [\ln|b-2| - \ln(2)] + \lim_{a \to 2^+} [\ln(1) - \ln|a-2|]$. 4) Evaluate the limits: Since $\lim_{b \to 2^-} \ln|b-2| = -\infty$ and $\lim_{a \to 2^+} \ln|a-2| = -\infty$, both integrals diverge. 5) The integral diverges.
How do you evaluate $\int_0^{\infty} \frac{x}{(1+x^2)^2} dx$?
1) Express as a limit: $\lim_{b \to \infty} \int_0^b \frac{x}{(1+x^2)^2} dx$. 2) Use u-substitution: Let $u = 1+x^2$, then $du = 2x dx$, so $x dx = \frac{1}{2} du$. The integral becomes $\frac{1}{2} \int \frac{1}{u^2} du = \frac{1}{2} \int u^{-2} du = \frac{1}{2} [-\frac{1}{u}] + C = -\frac{1}{2(1+x^2)} + C$. 3) Evaluate the integral: $\lim_{b \to \infty} [-\frac{1}{2(1+x^2)}]_0^b = \lim_{b \to \infty} [-\frac{1}{2(1+b^2)} - (-\frac{1}{2(1+0^2)})] = \lim_{b \to \infty} [-\frac{1}{2(1+b^2)} + \frac{1}{2}]$. 4) Evaluate the limit: $\lim_{b \to \infty} [-\frac{1}{2(1+b^2)} + \frac{1}{2}] = 0 + \frac{1}{2} = \frac{1}{2}$. 5) The integral converges to $\frac{1}{2}$.
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.
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 $e^x$?
$\int e^x dx = e^x + C$
