If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$.

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.

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 $\int_a^\infty g(x) dx$.

$\int_a^b F'(x) dx = F(b) - F(a)$

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.

$\int u dv = uv - \int v du$

$\int f(g(x))g'(x)dx = \int f(u)du$ where $u=g(x)$

Decompose a rational function into simpler fractions that are easier to integrate.

$\int \frac{1}{x} dx = \ln|x| + C$

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

Limits allow us to approach infinity without actually reaching it, enabling us to evaluate the integral's behavior.

Choose an arbitrary point 'c' and split the integral into two: one from negative infinity to 'c' and another from 'c' to infinity. Evaluate each separately.

Check if one or both limits are infinite, or if the function has a discontinuity within the integration interval.

Convergence indicates that the area under the curve approaches a finite value, allowing for a meaningful result.

Divergence indicates that the area under the curve does not approach a finite value, meaning the integral does not have a finite result.

It is used to evaluate the integral after expressing it as a limit, but before evaluating the limit itself.

Discontinuities can make an integral improper, requiring special treatment using limits.

Improper integrals can be used to find the area under a curve even when the curve extends to infinity or has discontinuities.

Substitution can simplify the integrand, making it easier to evaluate the integral before taking the limit.

Absolute value ensures that the logarithm is defined for all values within the integration interval, especially when dealing with negative values.

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.

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.

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.

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.

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.

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 $\pi$.

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 $\ln(2)$.

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.

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.

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 $\frac{1}{2}$.